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Buddhism and how Symmetry relates to it.

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on Thursday, 10 January 2013 in Academic and Scholars

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Symmetry

From Wikipedia, the free encyclopedia
 
 
Asymmetric (PSF).svg
Sphere symmetrical group o.
Leonardo da Vinci's Vitruvian Man (ca. 1487) is often used as a representation of symmetry in the human body and, by extension, the natural universe.
Symmetric arcades of a portico in the Great Mosque of Kairouanalso called the Mosque of Uqba, inTunisia.

Symmetry (from Greek συμμετρεῖν symmetría "measure together") generally conveys two primary meanings. The first is an imprecise sense of harmonious or aesthetically pleasing proportionality and balance;[1][2] such that it reflects beauty or perfection. The second meaning is a precise and well-defined concept of balance or "patterned self-similarity" that can be demonstrated or proved according to the rules of a formal system: by geometry, through physics or otherwise.

Although the meanings are distinguishable in some contexts, both meanings of "symmetry" are related and discussed in parallel.[2][3]

The precise notions of symmetry have various measures and operational definitions. For example, symmetry may be observed

This article describes these notions of symmetry from four perspectives. The first is that of symmetry in geometry, which is the most familiar type of symmetry for many people. The second perspective is the more general meaning of symmetry in mathematics as a whole. The third perspective describes symmetry as it relates to science and technology. In this context, symmetries underlie some of the most profound results found in modern physics, including aspects ofspace and time. Finally, a fourth perspective discusses symmetry in the humanities, covering its rich and varied use in historyarchitectureart, and religion.

The opposite of symmetry is asymmetry.

Contents

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[edit]In geometry

The most familiar type of symmetry for many people is geometrical symmetry. Formally, this means symmetry under a sub-group of the Euclidean group ofisometries in two or three dimensional Euclidean space. These isometries consist of reflections, rotations, translations and combinations of these basic operations.[7]

[edit]Reflection symmetry

Simetria-reflexion.svg
A butterfly with bilateral symmetry

Reflection symmetry, mirror symmetry, mirror-image symmetry, or bilateral symmetry is symmetry with respect to reflection.

In 1D, there is a point of symmetry. In 2D there is an axis of symmetry, in 3D a plane of symmetry. An object or figure which is indistinguishable from its transformed image is called mirror symmetric (see mirror image).

The axis of symmetry of a two-dimensional figure is a line such that, if a perpendicular is constructed, any two points lying on the perpendicular at equal distances from the axis of symmetry are identical. Another way to think about it is that if the shape were to be folded in half over the axis, the two halves would be identical: the two halves are each other's mirror image. Thus a square has four axes of symmetry, because there are four different ways to fold it and have the edges all match. A circle has infinitely many axes of symmetry, for the same reason.

If the letter T is reflected along a vertical axis, it appears the same. This is sometimes called vertical symmetry. One can better use an unambiguous formulation; e.g., "T has a vertical symmetry axis" or "T has left-right symmetry".

The triangles with this symmetry are isosceles, the quadrilaterals with this symmetry are the kites and the isosceles trapezoids.

For each line or plane of reflection, the symmetry group is isomorphic with Cs (see point groups in three dimensions), one of the three types of order two (involutions), hence algebraically C2. The fundamental domain is a half-plane or half-space.

Bilateria (bilateral animals, including humans) are more or less symmetric with respect to the sagittal plane.

[edit]Point reflection and other involutive isometries

Reflection symmetry can be generalized to other isometries of m-dimensional space which are involutions, such as

(x1, … xm) ↦ (−x1, … −xk, xk+1, … xm)

in certain system of Cartesian coordinates. This reflects the space along a mk-dimensional affine subspace. If k = m, then such transformation is known as point reflection, which on the plane(m = 2) is the same as the half-turn (180°) rotation; see below.

Such "reflection" keeps orientation if and only if k is even. This implies that for m = 3 (as well for other odd m) a point reflection changes orientation of the space, like mirror-image symmetry. That's why in physics the term P-symmetry is used for both point reflection and mirror symmetry (P stands for parity).

[edit]Rotational symmetry

Simetria-rotacion.svg

Rotational symmetry is symmetry with respect to some or all rotations in m-dimensional Euclidean space. Rotations are direct isometries; i.e., isometries preserving orientation. Therefore a symmetry group of rotational symmetry is a subgroup of E+(m).

Symmetry with respect to all rotations about all points implies translational symmetry with respect to all translations, and the symmetry group is the whole E+(m). This does not apply for objects because it makes space homogeneous, but it may apply for physical laws.

For symmetry with respect to rotations about a point we can take that point as origin. These rotations form the special orthogonal group SO(m), the group ofm × m orthogonal matrices with determinant 1. For m = 3 this is the rotation group SO(3).

In another meaning of the word, the rotation group of an object is the symmetry group within E+(m), the group of direct isometries; in other words, the intersection of the full symmetry group and the group of direct isometries. For chiral objects it is the same as the full symmetry group.

Laws of physics are SO(3)-invariant if they do not distinguish different directions in space. Because of Noether's theorem, rotational symmetry of a physical system is equivalent to the angular momentum conservation law. See also rotational invariance.

[edit]Translational symmetry

Translational symmetry leaves an object invariant under a discrete or continuous group of translations scriptstyle T_a(p) ;=; p ,+, a.

[edit]Glide reflection symmetry

Simetria-antitraslacional.svg

glide reflection symmetry (in 3D: a glide plane symmetry) means that a reflection in a line or plane combined with a translation along the line / in the plane, results in the same object. It implies translational symmetry with twice the translation vector. The symmetry group is isomorphic with Z.

[edit]Rotoreflection symmetry

Simetria-rotoreflexion.svg

In 3D, rotoreflection or improper rotation in the strict sense is rotation about an axis, combined with reflection in a plane perpendicular to that axis. As symmetry groups with regard to a roto-reflection we can distinguish:

  • the angle has no common divisor with 360°, the symmetry group is not discrete
  • 2n-fold rotoreflection (angle of 180°/n) with symmetry group S2n of order 2n (not to be confused with symmetric groups, for which the same notation is used; abstract group C2n); a special case is n = 1, inversion, because it does not depend on the axis and the plane, it is characterized by just the point of inversion.
  • Cnh (angle of 360°/n); for odd n this is generated by a single symmetry, and the abstract group is C2n, for even n this is not a basic symmetry but a combination. See also point groups in three dimensions.

[edit]Helical symmetry

Simetria-helicoidal.svg
A drill bit with helical symmetry.

Helical symmetry is the kind of symmetry seen in such everyday objects as springsSlinky toys, drill bits, and augers. It can be thought of as rotational symmetry along with translation along the axis of rotation, the screw axis. The concept of helical symmetry can be visualized as the tracing in three-dimensional space that results from rotating an object at an even angular speed while simultaneously moving at another even speed along its axis of rotation (translation). At any one point in time, these two motions combine to give a coiling angle that helps define the properties of the tracing. When the tracing object rotates quickly and translates slowly, the coiling angle will be close to 0°. Conversely, if the rotation is slow and the translation is speedy, the coiling angle will approach 90°.

Three main classes of helical symmetry can be distinguished based on the interplay of the angle of coiling and translation symmetries along the axis:

Infinite helical symmetry
If there are no distinguishing features along the length of a helix or helix-like object, the object will have infinite symmetry much like that of a circle, but with the additional requirement of translation along the long axis of the object to return it to its original appearance. A helix-like object is one that has at every point the regular angle of coiling of a helix, but which can also have a cross section of indefinitely high complexity, provided only that precisely the same cross section exists (usually after a rotation) at every point along the length of the object. Simple examples include evenly coiled springsslinkiesdrill bits, and augers. Stated more precisely, an object has infinite helical symmetries if for any small rotation of the object around its central axis there exists a point nearby (the translation distance) on that axis at which the object will appear exactly as it did before. It is this infinite helical symmetry that gives rise to the curious illusion of movement along the length of an auger or screw bit that is being rotated. It also provides the mechanically useful ability of such devices to move materials along their length, provided that they are combined with a force such as gravity or friction that allows the materials to resist simply rotating along with the drill or auger.
n-fold helical symmetry
If the requirement that every cross section of the helical object be identical is relaxed, additional lesser helical symmetries become possible. For example, the cross section of the helical object may change, but still repeats itself in a regular fashion along the axis of the helical object. Consequently, objects of this type will exhibit a symmetry after a rotation by some fixed angle θ and a translation by some fixed distance, but will not in general be invariant for any rotation angle. If the angle (rotation) at which the symmetry occurs divides evenly into a full circle (360°), the result is the helical equivalent of a regular polygon. This case is called n-fold helical symmetry, where n = 360°; e.g., double helix. This concept can be further generalized to include cases where scriptstyle mtheta is a multiple of 360° – that is, the cycle does eventually repeat, but only after more than one full rotation of the helical object.
Non-repeating helical symmetry
This is the case in which the angle of rotation θ required to observe the symmetry is irrational. The angle of rotation never repeats exactly no matter how many times the helix is rotated. Such symmetries are created by using a non-repeating point group in two dimensionsDNA is an example of this type of non-repeating helical symmetry.[citation needed]

[edit]Non-isometric symmetries

A wider definition of geometric symmetry allows operations from a larger group than the Euclidean group of isometries. Examples of larger geometric symmetry groups are:

In Felix Klein's Erlangen program, each possible group of symmetries defines a geometry in which objects that are related by a member of the symmetry group are considered to be equivalent. For example, the Euclidean group defines Euclidean geometry, whereas the group of Möbius transformations defines projective geometry.

[edit]Scale symmetry and fractals

Scale symmetry refers to the idea that if an object is expanded or reduced in size, the new object has the same properties as the original. Scale symmetry is notable for the fact that it does notexist for most physical systems, a point that was first discerned by Galileo. Simple examples of the lack of scale symmetry in the physical world include the difference in the strength and size of the legs of elephants versus mice, and the observation that if a candle made of soft wax was enlarged to the size of a tall tree, it would immediately collapse under its own weight.

A more subtle form of scale symmetry is demonstrated by fractals. As conceived by Benoît Mandelbrot, fractals are a mathematical concept in which the structure of a complex form looks similar or even exactly the same no matter what degree of magnification is used to examine it. A coast is an example of a naturally occurring fractal, since it retains roughly comparable and similar-appearing complexity at every level from the view of a satellite to a microscopic examination of how the water laps up against individual grains of sand. The branching of trees, which enables children to use small twigs as stand-ins for full trees in dioramas, is another example.

This similarity to naturally occurring phenomena provides fractals with an everyday familiarity not typically seen with mathematically generated functions. As a consequence, they can produce strikingly beautiful results such as the Mandelbrot set. Intriguingly, fractals have also found a place in CG, or computer-generated movie effects, where their ability to create very complex curves with fractal symmetries results in more realistic virtual worlds.

[edit]In mathematics

In formal terms, we say that a mathematical object is symmetric with respect to a given mathematical operation, if, when applied to the object, this operation preserves some property of the object. The set of operations that preserve a given property of the object form a group. Two objects are symmetric to each other with respect to a given group of operations if one is obtained from the other by some of the operations (and vice versa).

[edit]Mathematical model for symmetry

The set of all symmetry operations considered, on all objects in a set X, can be modeled as a group action g : G × X → X, where the image of g in G and x in X is written as g·x. If, for some gg·xy then x and y are said to be symmetrical to each other. For each object x, operations g for which g·x = x form a group, the symmetry group of the object, a subgroup of G. If the symmetry group of x is the trivial group then x is said to be asymmetric, otherwise symmetric.

A general example is that G is a group of bijections gV → V acting on the set of functions xV → W by (gx)(v) = x[g−1(v)] (or a restricted set of such functions that is closed under the group action). Thus a group of bijections of space induces a group action on "objects" in it. The symmetry group of x consists of all g for which x(v) = x[g(v)] for all vG is the symmetry group of the space itself, and of any object that is uniform throughout space. Some subgroups of G may not be the symmetry group of any object. For example, if the group contains for every v and w in V a gsuch that g(v) = w, then only the symmetry groups of constant functions x contain that group. However, the symmetry group of constant functions is G itself.

In a modified version for vector fields, we have (gx)(v) = h(gx[g−1(v)]) where h rotates any vectors and pseudovectors in x, and inverts any vectors (but not pseudovectors) according to rotation and inversion in g, see symmetry in physics. The symmetry group of x consists of all g for which x(v) = h(gx[g(v)]) for all v. In this case the symmetry group of a constant function may be a proper subgroup of G: a constant vector has only rotational symmetry with respect to rotation about an axis if that axis is in the direction of the vector, and only inversion symmetry if it is zero.

For a common notion of symmetry in Euclidean spaceG is the Euclidean group E(n), the group of isometries, and V is the Euclidean space. The rotation group of an object is the symmetry group if G is restricted to E+(n), the group of direct isometries. (For generalizations, see the next subsection.) Objects can be modeled as functions x, of which a value may represent a selection of properties such as color, density, chemical composition, etc. Depending on the selection we consider just symmetries of sets of points (x is just a Boolean function of position v), or, at the other extreme; e.g., symmetry of right and left hand with all their structure.

For a given symmetry group, the properties of part of the object, fully define the whole object. Considering points equivalent which, due to the symmetry, have the same properties, the equivalence classes are the orbits of the group action on the space itself. We need the value of x at one point in every orbit to define the full object. A set of such representatives forms a fundamental domain. The smallest fundamental domain does not have a symmetry; in this sense, one can say that symmetry relies upon asymmetry.

An object with a desired symmetry can be produced by choosing for every orbit a single function value. Starting from a given object x we can, e.g.:

  • Take the values in a fundamental domain (i.e., add copies of the object).
  • Take for each orbit some kind of average or sum of the values of x at the points of the orbit (ditto, where the copies may overlap).

If it is desired to have no more symmetry than that in the symmetry group, then the object to be copied should be asymmetric.

As pointed out above, some groups of isometries are not the symmetry group of any object, except in the modified model for vector fields. For example, this applies in 1D for the group of all translations. The fundamental domain is only one point, so we can not make it asymmetric, so any "pattern" invariant under translation is also invariant under reflection (these are the uniform "patterns").

In the vector field version continuous translational symmetry does not imply reflectional symmetry: the function value is constant, but if it contains nonzero vectors, there is no reflectional symmetry. If there is also reflectional symmetry, the constant function value contains no nonzero vectors, but it may contain nonzero pseudovectors. A corresponding 3D example is an infinitecylinder with a current perpendicular to the axis; the magnetic field (a pseudovector) is, in the direction of the cylinder, constant, but nonzero. For vectors (in particular the current density) we have symmetry in every plane perpendicular to the cylinder, as well as cylindrical symmetry. This cylindrical symmetry without mirror planes through the axis is also only possible in the vector field version of the symmetry concept. A similar example is a cylinder rotating about its axis, where magnetic field and current density are replaced by angular momentum and velocity, respectively.

A symmetry group is said to act transitively on a repeated feature of an object if, for every pair of occurrences of the feature there is a symmetry operation mapping the first to the second. For example, in 1D, the symmetry group of {…, 1, 2, 5, 6, 9, 10, 13, 14, …} acts transitively on all these points, while {…, 1, 2, 3, 5, 6, 7, 9, 10, 11, 13, 14, 15, …} does not act transitively on all points. Equivalently, the first set is only one conjugacy class with respect to isometries, while the second has two classes.

[edit]Symmetric functions

A symmetric function is a function which is unchanged by any permutation of its variables. For example, x + y + z and xy + yz + xz are symmetric functions, whereas x2 – yz is not.

A function may be unchanged by a sub-group of all the permutations of its variables. For example, ac + 3ab + bc is unchanged if a and b are exchanged; its symmetry group is isomorphic to C2.

[edit]Symmetry in logic

dyadic relation R is symmetric if and only if, whenever it's true that Rab, it's true that Rba. Thus, "is the same age as" is symmetrical, for if Paul is the same age as Mary, then Mary is the same age as Paul.

Symmetric binary logical connectives are and (∧, or &), or (∨, or |), biconditional (if and only if) (↔), nand (not-and, or ⊼), xor (not-biconditional, or ⊻), and nor (not-or, or ⊽).

[edit]In science

[edit]Symmetry in physics

Symmetry in physics has been generalized to mean invariance—that is, lack of change—under any kind of transformation, for example arbitrary coordinate transformations. This concept has become one of the most powerful tools of theoretical physics, as it has become evident that practically all laws of nature originate in symmetries. In fact, this role inspired the Nobel laureate PW Anderson to write in his widely read 1972 article More is Different that "it is only slightly overstating the case to say that physics is the study of symmetry." See Noether's theorem (which, in greatly simplified form, states that for every continuous mathematical symmetry, there is a corresponding conserved quantity; a conserved current, in Noether's original language); and also,Wigner's classification, which says that the symmetries of the laws of physics determine the properties of the particles found in nature.

[edit]Symmetry in physical objects

[edit]Classical objects

Although an everyday object may appear exactly the same after a symmetry operation such as a rotation or an exchange of two identical parts has been performed on it, it is readily apparent that such a symmetry is true only as an approximation for any ordinary physical object.

For example, if one rotates a precisely machined aluminum equilateral triangle 120 degrees around its center, a casual observer brought in before and after the rotation will be unable to decide whether or not such a rotation took place. However, the reality is that each corner of a triangle will always appear unique when examined with sufficient precision. An observer armed with sufficiently detailed measuring equipment such as optical or electron microscopes will not be fooled; he will immediately recognize that the object has been rotated by looking for details such ascrystals or minor deformities.

Such simple thought experiments show that assertions of symmetry in everyday physical objects are always a matter of approximate similarity rather than of precise mathematical sameness. The most important consequence of this approximate nature of symmetries in everyday physical objects is that such symmetries have minimal or no impacts on the physics of such objects. Consequently, only the deeper symmetries of space and time play a major role in classical physics; that is, the physics of large, everyday objects.

[edit]Quantum objects

Remarkably, there exists a realm of physics for which mathematical assertions of simple symmetries in real objects cease to be approximations. That is the domain of quantum physics, which for the most part is the physics of very small, very simple objects such as electronsprotonslight, and atoms.

Unlike everyday objects, objects such as electrons have very limited numbers of configurations, called states, in which they can exist. This means that when symmetry operations such as exchanging the positions of components are applied to them, the resulting new configurations often cannot be distinguished from the originals no matter how diligent an observer is. Consequently, for sufficiently small and simple objects the generic mathematical symmetry assertion F(x) = x ceases to be approximate, and instead becomes an experimentally precise and accurate description of the situation in the real world.

[edit]Consequences of quantum symmetry

While it makes sense that symmetries could become exact when applied to very simple objects, the immediate intuition is that such a detail should not affect the physics of such objects in any significant way. This is in part because it is very difficult to view the concept of exact similarity as physically meaningful. Our mental picture of such situations is invariably the same one we use for large objects: We picture objects or configurations that are very, very similar, but for which if we could "look closer" we would still be able to tell the difference.

However, the assumption that exact symmetries in very small objects should not make any difference in their physics was discovered in the early 1900s to be spectacularly incorrect. The situation was succinctly summarized by Richard Feynman in the direct transcripts of his Feynman Lectures on Physics, Volume III, Section 3.4, Identical particles. (Unfortunately, the quote was edited out of the printed version of the same lecture.)

… if there is a physical situation in which it is impossible to tell which way it happened, it always interferes; it never fails.

The word "interferes" in this context is a quick way of saying that such objects fall under the rules of quantum mechanics, in which they behave more like waves that interfere than like everyday large objects.

In short, when an object becomes so simple that a symmetry assertion of the form F(x) = x becomes an exact statement of experimentally verifiable sameness, x ceases to follow the rules ofclassical physics and must instead be modeled using the more complex, and often far less intuitive, rules of quantum physics.

This transition also provides an important insight into why the mathematics of symmetry are so deeply intertwined with those of quantum mechanics. When physical systems make the transition from symmetries that are approximate to ones that are exact, the mathematical expressions of those symmetries cease to be approximations and are transformed into precise definitions of the underlying nature of the objects. From that point on, the correlation of such objects to their mathematical descriptions becomes so close that it is difficult to separate the two.

[edit]Generalizations of symmetry

If we have a given set of objects with some structure, then it is possible for a symmetry to merely convert only one object into another, instead of acting upon all possible objects simultaneously. This requires a generalization from the concept of symmetry group to that of a groupoid. Indeed, A. Connes in his book "Non-commutative geometry" writes that Heisenberg discovered quantum mechanics by considering the groupoid of transitions of the hydrogen spectrum.

The notion of groupoid also leads to notions of multiple groupoids, namely sets with many compatible groupoid structures, a structure which trivialises to abelian groups if one restricts to groups. This leads to prospects of higher order symmetry which have been a little explored, as follows.

The automorphisms of a set, or a set with some structure, form a group, which models a homotopy 1-type. The automorphisms of a group G naturally form a crossed module scriptstyle G ;to; mathrm{Aut}(G), and crossed modules give an algebraic model of homotopy 2-types. At the next stage, automorphisms of a crossed module fit into a structure known as a crossed square, and this structure is known to give an algebraic model of homotopy 3-types. It is not known how this procedure of generalising symmetry may be continued, although crossed n-cubes have been defined and used in algebraic topology, and these structures are only slowly being brought into theoretical physics.[7][8]

Physicists have come up with other directions of generalization, such as supersymmetry and quantum groups, yet the different options are indistinguishable during various circumstances.

[edit]Symmetry in biology

[edit]Symmetry in chemistry

Symmetry is important to chemistry because it explains observations in spectroscopyquantum chemistry and crystallography. It draws heavily on group theory.

[edit]In history, religion, and culture

In any human endeavor for which an impressive visual result is part of the desired objective, symmetries play a profound role. The innate appeal of symmetry can be found in our reactions to happening across highly symmetrical natural objects, such as precisely formed crystals or beautifully spiraled seashells. Our first reaction in finding such an object often is to wonder whether we have found an object created by a fellow human, followed quickly by surprise that the symmetries that caught our attention are derived from nature itself. In both reactions we give away our inclination to view symmetries both as beautiful and, in some fashion, informative of the world around us.[citation needed]

[edit]Symmetry in social interactions

People observe the symmetrical nature, often including asymmetrical balance, of social interactions in a variety of contexts. These include assessments of reciprocity, empathy, apology, dialog, respect, justice, and revenge. Symmetrical interactions send the message "we are all the same" while asymmetrical interactions send the message "I am special; better than you." Peer relationships are based on symmetry, power relationships are based on asymmetry.[9]

[edit]Symmetry in architecture

The ceiling of Lotfollah mosqueIsfahanIranhas rotational symmetry of order eight and eight lines of reflection.
Leaning Tower of Pisa
The Taj Mahal has bilateral symmetry.

Another human endeavor in which the visual result plays a vital part in the overall result is architecture. Both in ancient and modern times, the ability of a large structure to impress or even intimidate its viewers has often been a major part of its purpose, and the use of symmetry is an inescapable aspect of how to accomplish such goals.

Just a few examples of ancient architectures that made powerful use of symmetry to impress those around them included the Egyptian Pyramids, the Greek Parthenon, the first and second Temple of Jerusalem, China'sForbidden CityCambodia's Angkor Wat complex, and the many temples and pyramids of ancient Pre-Columbian civilizations. More recent historical examples of architectures emphasizing symmetries includeGothic architecture cathedrals, and American President Thomas Jefferson's Monticello home. The Taj Mahal is also an example of symmetry.[10]

An interesting example of a broken symmetry in architecture is the Leaning Tower of Pisa, whose notoriety stems in no small part not for the intended symmetry of its design, but for the violation of that symmetry from the lean that developed while it was still under construction. Modern examples of architectures that make impressive or complex use of various symmetries include Australia's Sydney Opera House and Houston, Texas's simplerAstrodome.

Symmetry finds its ways into architecture at every scale, from the overall external views, through the layout of the individual floor plans, and down to the design of individual building elements such as intricately carved doors, stained glass windowstile mosaicsfriezes, stairwells, stair rails, and balustrades. For sheer complexity and sophistication in the exploitation of symmetry as an architectural element, Islamic buildings such as the Taj Mahal often eclipse those of other cultures and ages, due in part to the general prohibition of Islam against using images of people or animals.[11][12]

[edit]Symmetry in pottery and metal vessels

Persian vessel (4th millennium BC)

Since the earliest uses of pottery wheels to help shape clay vessels, pottery has had a strong relationship to symmetry. As a minimum, pottery created using a wheel necessarily begins with full rotational symmetry in its cross-section, while allowing substantial freedom of shape in the vertical direction. Upon this inherently symmetrical starting point cultures from ancient times have tended to add further patterns that tend to exploit or in many cases reduce the original full rotational symmetry to a point where some specific visual objective is achieved. For example, Persian pottery dating from the fourth millennium BC and earlier used symmetric zigzags, squares, cross-hatchings, and repetitions of figures to produce more complex and visually striking overall designs.

Cast metal vessels lacked the inherent rotational symmetry of wheel-made pottery, but otherwise provided a similar opportunity to decorate their surfaces with patterns pleasing to those who used them. The ancient Chinese, for example, used symmetrical patterns in their bronze castings as early as the 17th century BC. Bronze vessels exhibited both a bilateral main motif and a repetitive translated border design.[13][14][15]

[edit]Symmetry in quilts

Kitchen Kaleidoscope Block

As quilts are made from square blocks (usually 9, 16, or 25 pieces to a block) with each smaller piece usually consisting of fabric triangles, the craft lends itself readily to the application of symmetry.[16]

[edit]Symmetry in carpets and rugs

Persian rug.

A long tradition of the use of symmetry in carpet and rug patterns spans a variety of cultures. American Navajo Indians used bold diagonals and rectangular motifs. Many Oriental rugs have intricate reflected centers and borders that translate a pattern. Not surprisingly, rectangular rugs typically use quadrilateral symmetry—that is, motifs that are reflected across both the horizontal and vertical axes.[17][18]

[edit]Symmetry in music

Major and minor triads on the white piano keys are symmetrical to the D. (compare article) (file)

Symmetry is not restricted to the visual arts. Its role in the history of music touches many aspects of the creation and perception of music.

[edit]Musical form

Symmetry has been used as a formal constraint by many composers, such as the arch (swell) form (ABCBA) used by Steve ReichBéla Bartók, and James Tenney. In classical music, Bach used the symmetry concepts of permutation and invariance.[19]

[edit]Pitch structures

Symmetry is also an important consideration in the formation of scales and chords, traditional or tonal music being made up of non-symmetrical groups of pitches, such as the diatonic scale or the major chordSymmetrical scales or chords, such as the whole tone scaleaugmented chord, or diminished seventh chord (diminished-diminished seventh), are said to lack direction or a sense of forward motion, are ambiguous as to the key or tonal center, and have a less specific diatonic functionality. However, composers such as Alban BergBéla Bartók, and George Perle have used axes of symmetry and/or interval cycles in an analogous way to keys or non-tonal tonalcenters.

Perle (1992) explains "C–E, D–F♯, [and] Eb–G, are different instances of the same interval … the other kind of identity. … has to do with axes of symmetry. C–E belongs to a family of symmetrically related dyads as follows:"

D   D♯   E   F   F♯   G   G♯
D   C♯   C   B   A♯   A   G♯

Thus in addition to being part of the interval-4 family, C–E is also a part of the sum-4 family (with C equal to 0).

+ 2   3   4   5   6   7   8
2   1   0   11   10   9   8
4   4   4   4   4   4   4

Interval cycles are symmetrical and thus non-diatonic. However, a seven pitch segment of C5 (the cycle of fifths, which are enharmonic with the cycle of fourths) will produce the diatonic major scale. Cyclic tonal progressions in the works of Romantic composers such as Gustav Mahler and Richard Wagner form a link with the cyclic pitch successions in the atonal music of Modernists such as Bartók, Alexander ScriabinEdgard Varèse, and the Vienna school. At the same time, these progressions signal the end of tonality.

The first extended composition consistently based on symmetrical pitch relations was probably Alban Berg's Quartet, Op. 3 (1910). (Perle, 1990)

[edit]Equivalency

Tone rows or pitch class sets which are invariant under retrograde are horizontally symmetrical, under inversion vertically. See also Asymmetric rhythm.

[edit]Symmetry in other arts and crafts

The concept of symmetry is applied to the design of objects of all shapes and sizes. Other examples include beadworkfurnituresand paintings,knotworkmasksmusical instruments, and many other endeavors.

[edit]Symmetry in aesthetics

The relationship of symmetry to aesthetics is complex. Certain simple symmetries, and in particular bilateral symmetry, seem to be deeply ingrained in the inherent perception by humans of the likely health or fitness of other living creatures, as can be seen by the simple experiment of distorting one side of the image of an attractive face and asking viewers to rate the attractiveness of the resulting image. Consequently, such symmetries that mimic biology tend to have an innate appeal that in turn drives a powerful tendency to create artifacts with similar symmetry. One only needs to imagine the difficulty in trying to market a highly asymmetrical car or truck to general automotive buyers to understand the power of biologically inspired symmetries such as bilateral symmetry.

Another more subtle appeal of symmetry is that of simplicity, which in turn has an implication of safety, security, and familiarity.[citation needed] A highly symmetrical room, for example, is unavoidably also a room in which anything out of place or potentially threatening can be identified easily and quickly.[citation needed] For example, people who have grown up in houses full of exact right angles and precisely identical artifacts can find their first experience in staying in a room with no exact right angles and no exactly identical artifacts to be highly disquieting.[citation needed]Symmetry thus can be a source of comfort not only as an indicator of biological health, but also of a safe and well-understood living environment.

Opposed to this is the tendency for excessive symmetry to be perceived as boring or uninteresting. Humans in particular have a powerful desire to exploit new opportunities or explore new possibilities, and an excessive degree of symmetry can convey a lack of such opportunities.[citation needed] Most people display a preference for figures that have a certain degree of simplicity and symmetry, but enough complexity to make them interesting.[20]

Yet another possibility is that when symmetries become too complex or too challenging, the human mind has a tendency to "tune them out" and perceive them in yet another fashion: as noisethat conveys no useful information.[citation needed]

Finally, perceptions and appreciation of symmetries are also dependent on cultural background. The far greater use of complex geometric symmetries in many Islamic cultures, for example, makes it more likely that people from such cultures will appreciate such art forms (or, conversely, to rebel against them).[citation needed]

As in many human endeavors, the result of the confluence of many such factors is that effective use of symmetry in art and architecture is complex, intuitive, and highly dependent on the skills of the individuals who must weave and combine such factors within their own creative work. Along with texture, color, proportion, and other factors, symmetry is a powerful ingredient in any such synthesis; one only need to examine the Taj Mahal to powerful role that symmetry plays in determining the aesthetic appeal of an object.

Modernist architecture rejects symmetry, stating only a bad architect relies on symmetry;[citation needed] instead of symmetrical layout of blocks, masses and structures, Modernist architecture relies on wings and balance of masses. This notion of getting rid of symmetry was first encountered in International style. Some people find asymmetrical layouts of buildings and structures revolutionizing; other find them restless, boring and unnatural.

A few examples of the more explicit use of symmetries in art can be found in the remarkable art of M.C. Escher, the creative design of the mathematical concept of a wallpaper group, and the many applications (both mathematical and real world) of tiling.

[edit]See also

[edit]References

  1. ^ Penrose, Roger (2007). Fearful Symmetry. City: Princeton. ISBN 978-0-691-13482-6.
  2. a b For example, Aristotle ascribed spherical shape to the heavenly bodies, attributing this formally defined geometric measure of symmetry to the natural order and perfection of the cosmos.
  3. ^ Weyl 1982
  4. ^ For example, operations such as moving across a regularly patterned tile floor or rotating an eight-sided vase, or complex transformations of equations or in the way music is played.
  5. ^ See, e.g., Mainzer, Klaus (2005). Symmetry And Complexity: The Spirit and Beauty of Nonlinear Science. World Scientific. ISBN 981-256-192-7.
  6. ^ Symmetric objects can be material, such as a person, crystalquiltfloor tiles, or molecule, or it can be an abstract structure such as a mathematical equation or a series of tones (music).
  7. a b `Higher dimensional group theory'
  8. ^ n-category cafe – discussion of n-groups
  9. ^ Emotional Competency Entry describing Symmetry
  10. ^ Gregory Neil Derry (2002), What Science Is and How It Works, Princeton University Press, p. 269
  11. ^ Williams: Symmetry in Architecture
  12. ^ Aslaksen: Mathematics in Art and Architecture
  13. ^ Chinavoc: The Art of Chinese Bronzes
  14. ^ Grant: Iranian Pottery in the Oriental Institute
  15. ^ The Metropolitan Museum of Art – Islamic Art
  16. ^ Quate: Exploring Geometry Through Quilts
  17. ^ Mallet: Tribal Oriental Rugs
  18. ^ Dilucchio: Navajo Rugs
  19. ^ see ("Fugue No. 21," pdf or Shockwave)
  20. ^ Arnheim, Rudolf (1969). Visual Thinking. University of California Press.

[edit]External links

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Symmetry
From Wikipedia, the free encyclopedia
For other uses, see Symmetry (disambiguation).




Sphere symmetrical group o.


Leonardo da Vinci's Vitruvian Man (ca. 1487) is often used as a representation of symmetry in the human body and, by extension, the natural universe.


Symmetric arcades of a portico in the Great Mosque of Kairouan also called the Mosque of Uqba, in Tunisia.
Symmetry (from Greek συμμετρεῖν symmetría "measure together") generally conveys two primary meanings. The first is an imprecise sense of harmonious or aesthetically pleasing proportionality and balance;[1][2] such that it reflects beauty or perfection. The second meaning is a precise and well-defined concept of balance or "patterned self-similarity" that can be demonstrated or proved according to the rules of a formal system: by geometry, through physics or otherwise.
Although the meanings are distinguishable in some contexts, both meanings of "symmetry" are related and discussed in parallel.[2][3]
The precise notions of symmetry have various measures and operational definitions. For example, symmetry may be observed
with respect to the passage of time;
as a spatial relationship;
through geometric transformations such as scaling, reflection, and rotation;
through other kinds of functional transformations;[4] and
as an aspect of abstract objects, theoretic models, language, music and even knowledge itself.[5][6]
This article describes these notions of symmetry from four perspectives. The first is that of symmetry in geometry, which is the most familiar type of symmetry for many people. The second perspective is the more general meaning of symmetry in mathematics as a whole. The third perspective describes symmetry as it relates to science and technology. In this context, symmetries underlie some of the most profound results found in modern physics, including aspects of space and time. Finally, a fourth perspective discusses symmetry in the humanities, covering its rich and varied use in history, architecture, art, and religion.
The opposite of symmetry is asymmetry.
Contents [hide]
1 In geometry
1.1 Reflection symmetry
1.2 Point reflection and other involutive isometries
1.3 Rotational symmetry
1.4 Translational symmetry
1.5 Glide reflection symmetry
1.6 Rotoreflection symmetry
1.7 Helical symmetry
1.8 Non-isometric symmetries
1.9 Scale symmetry and fractals
2 In mathematics
2.1 Mathematical model for symmetry
2.2 Symmetric functions
2.3 Symmetry in logic
3 In science
3.1 Symmetry in physics
3.2 Symmetry in physical objects
3.2.1 Classical objects
3.2.2 Quantum objects
3.2.3 Consequences of quantum symmetry
3.3 Generalizations of symmetry
3.4 Symmetry in biology
3.5 Symmetry in chemistry
4 In history, religion, and culture
4.1 Symmetry in social interactions
4.2 Symmetry in architecture
4.3 Symmetry in pottery and metal vessels
4.4 Symmetry in quilts
4.5 Symmetry in carpets and rugs
4.6 Symmetry in music
4.6.1 Musical form
4.6.2 Pitch structures
4.6.3 Equivalency
4.7 Symmetry in other arts and crafts
4.8 Symmetry in aesthetics
5 See also
6 References
7 External links
[edit]In geometry

The most familiar type of symmetry for many people is geometrical symmetry. Formally, this means symmetry under a sub-group of the Euclidean group of isometries in two or three dimensional Euclidean space. These isometries consist of reflections, rotations, translations and combinations of these basic operations.[7]
[edit]Reflection symmetry
Main article: reflection symmetry



A butterfly with bilateral symmetry
Reflection symmetry, mirror symmetry, mirror-image symmetry, or bilateral symmetry is symmetry with respect to reflection.
In 1D, there is a point of symmetry. In 2D there is an axis of symmetry, in 3D a plane of symmetry. An object or figure which is indistinguishable from its transformed image is called mirror symmetric (see mirror image).
The axis of symmetry of a two-dimensional figure is a line such that, if a perpendicular is constructed, any two points lying on the perpendicular at equal distances from the axis of symmetry are identical. Another way to think about it is that if the shape were to be folded in half over the axis, the two halves would be identical: the two halves are each other's mirror image. Thus a square has four axes of symmetry, because there are four different ways to fold it and have the edges all match. A circle has infinitely many axes of symmetry, for the same reason.
If the letter T is reflected along a vertical axis, it appears the same. This is sometimes called vertical symmetry. One can better use an unambiguous formulation; e.g., "T has a vertical symmetry axis" or "T has left-right symmetry".
The triangles with this symmetry are isosceles, the quadrilaterals with this symmetry are the kites and the isosceles trapezoids.
For each line or plane of reflection, the symmetry group is isomorphic with Cs (see point groups in three dimensions), one of the three types of order two (involutions), hence algebraically C2. The fundamental domain is a half-plane or half-space.
Bilateria (bilateral animals, including humans) are more or less symmetric with respect to the sagittal plane.
[edit]Point reflection and other involutive isometries
Reflection symmetry can be generalized to other isometries of m-dimensional space which are involutions, such as
(x1, … xm) ↦ (−x1, … −xk, xk+1, … xm)
in certain system of Cartesian coordinates. This reflects the space along a m−k-dimensional affine subspace. If k = m, then such transformation is known as point reflection, which on the plane (m = 2) is the same as the half-turn (180°) rotation; see below.
Such "reflection" keeps orientation if and only if k is even. This implies that for m = 3 (as well for other odd m) a point reflection changes orientation of the space, like mirror-image symmetry. That's why in physics the term P-symmetry is used for both point reflection and mirror symmetry (P stands for parity).
[edit]Rotational symmetry

Main article: rotational symmetry
Rotational symmetry is symmetry with respect to some or all rotations in m-dimensional Euclidean space. Rotations are direct isometries; i.e., isometries preserving orientation. Therefore a symmetry group of rotational symmetry is a subgroup of E+(m).
Symmetry with respect to all rotations about all points implies translational symmetry with respect to all translations, and the symmetry group is the whole E+(m). This does not apply for objects because it makes space homogeneous, but it may apply for physical laws.
For symmetry with respect to rotations about a point we can take that point as origin. These rotations form the special orthogonal group SO(m), the group of m × m orthogonal matrices with determinant 1. For m = 3 this is the rotation group SO(3).
In another meaning of the word, the rotation group of an object is the symmetry group within E+(m), the group of direct isometries; in other words, the intersection of the full symmetry group and the group of direct isometries. For chiral objects it is the same as the full symmetry group.
Laws of physics are SO(3)-invariant if they do not distinguish different directions in space. Because of Noether's theorem, rotational symmetry of a physical system is equivalent to the angular momentum conservation law. See also rotational invariance.
[edit]Translational symmetry
Main article: Translational symmetry
Translational symmetry leaves an object invariant under a discrete or continuous group of translations .
[edit]Glide reflection symmetry

A glide reflection symmetry (in 3D: a glide plane symmetry) means that a reflection in a line or plane combined with a translation along the line / in the plane, results in the same object. It implies translational symmetry with twice the translation vector. The symmetry group is isomorphic with Z.
[edit]Rotoreflection symmetry

In 3D, rotoreflection or improper rotation in the strict sense is rotation about an axis, combined with reflection in a plane perpendicular to that axis. As symmetry groups with regard to a roto-reflection we can distinguish:
the angle has no common divisor with 360°, the symmetry group is not discrete
2n-fold rotoreflection (angle of 180°/n) with symmetry group S2n of order 2n (not to be confused with symmetric groups, for which the same notation is used; abstract group C2n); a special case is n = 1, inversion, because it does not depend on the axis and the plane, it is characterized by just the point of inversion.
Cnh (angle of 360°/n); for odd n this is generated by a single symmetry, and the abstract group is C2n, for even n this is not a basic symmetry but a combination. See also point groups in three dimensions.
[edit]Helical symmetry



A drill bit with helical symmetry.
See also: Screw axis
Helical symmetry is the kind of symmetry seen in such everyday objects as springs, Slinky toys, drill bits, and augers. It can be thought of as rotational symmetry along with translation along the axis of rotation, the screw axis. The concept of helical symmetry can be visualized as the tracing in three-dimensional space that results from rotating an object at an even angular speed while simultaneously moving at another even speed along its axis of rotation (translation). At any one point in time, these two motions combine to give a coiling angle that helps define the properties of the tracing. When the tracing object rotates quickly and translates slowly, the coiling angle will be close to 0°. Conversely, if the rotation is slow and the translation is speedy, the coiling angle will approach 90°.
Three main classes of helical symmetry can be distinguished based on the interplay of the angle of coiling and translation symmetries along the axis:
Infinite helical symmetry
If there are no distinguishing features along the length of a helix or helix-like object, the object will have infinite symmetry much like that of a circle, but with the additional requirement of translation along the long axis of the object to return it to its original appearance. A helix-like object is one that has at every point the regular angle of coiling of a helix, but which can also have a cross section of indefinitely high complexity, provided only that precisely the same cross section exists (usually after a rotation) at every point along the length of the object. Simple examples include evenly coiled springs, slinkies, drill bits, and augers. Stated more precisely, an object has infinite helical symmetries if for any small rotation of the object around its central axis there exists a point nearby (the translation distance) on that axis at which the object will appear exactly as it did before. It is this infinite helical symmetry that gives rise to the curious illusion of movement along the length of an auger or screw bit that is being rotated. It also provides the mechanically useful ability of such devices to move materials along their length, provided that they are combined with a force such as gravity or friction that allows the materials to resist simply rotating along with the drill or auger.
n-fold helical symmetry
If the requirement that every cross section of the helical object be identical is relaxed, additional lesser helical symmetries become possible. For example, the cross section of the helical object may change, but still repeats itself in a regular fashion along the axis of the helical object. Consequently, objects of this type will exhibit a symmetry after a rotation by some fixed angle θ and a translation by some fixed distance, but will not in general be invariant for any rotation angle. If the angle (rotation) at which the symmetry occurs divides evenly into a full circle (360°), the result is the helical equivalent of a regular polygon. This case is called n-fold helical symmetry, where n = 360°; e.g., double helix. This concept can be further generalized to include cases where is a multiple of 360° – that is, the cycle does eventually repeat, but only after more than one full rotation of the helical object.
Non-repeating helical symmetry
This is the case in which the angle of rotation θ required to observe the symmetry is irrational. The angle of rotation never repeats exactly no matter how many times the helix is rotated. Such symmetries are created by using a non-repeating point group in two dimensions. DNA is an example of this type of non-repeating helical symmetry.[citation needed]
[edit]Non-isometric symmetries
A wider definition of geometric symmetry allows operations from a larger group than the Euclidean group of isometries. Examples of larger geometric symmetry groups are:
The group of similarity transformations; i.e., affine transformations represented by a matrix A that is a scalar times an orthogonal matrix. Thus homothety is added, self-similarity is considered a symmetry.
The group of affine transformations represented by a matrix A with determinant 1 or −1; i.e., the transformations which preserve area.
This adds, e.g., oblique reflection symmetry.
The group of all bijective affine transformations.
The group of Möbius transformations which preserve cross-ratios.
This adds, e.g., inversive reflections such as circle reflection on the plane.
In Felix Klein's Erlangen program, each possible group of symmetries defines a geometry in which objects that are related by a member of the symmetry group are considered to be equivalent. For example, the Euclidean group defines Euclidean geometry, whereas the group of Möbius transformations defines projective geometry.
[edit]Scale symmetry and fractals
Scale symmetry refers to the idea that if an object is expanded or reduced in size, the new object has the same properties as the original. Scale symmetry is notable for the fact that it does not exist for most physical systems, a point that was first discerned by Galileo. Simple examples of the lack of scale symmetry in the physical world include the difference in the strength and size of the legs of elephants versus mice, and the observation that if a candle made of soft wax was enlarged to the size of a tall tree, it would immediately collapse under its own weight.
A more subtle form of scale symmetry is demonstrated by fractals. As conceived by Benoît Mandelbrot, fractals are a mathematical concept in which the structure of a complex form looks similar or even exactly the same no matter what degree of magnification is used to examine it. A coast is an example of a naturally occurring fractal, since it retains roughly comparable and similar-appearing complexity at every level from the view of a satellite to a microscopic examination of how the water laps up against individual grains of sand. The branching of trees, which enables children to use small twigs as stand-ins for full trees in dioramas, is another example.
This similarity to naturally occurring phenomena provides fractals with an everyday familiarity not typically seen with mathematically generated functions. As a consequence, they can produce strikingly beautiful results such as the Mandelbrot set. Intriguingly, fractals have also found a place in CG, or computer-generated movie effects, where their ability to create very complex curves with fractal symmetries results in more realistic virtual worlds.
[edit]In mathematics

Main article: Symmetry in mathematics
In formal terms, we say that a mathematical object is symmetric with respect to a given mathematical operation, if, when applied to the object, this operation preserves some property of the object. The set of operations that preserve a given property of the object form a group. Two objects are symmetric to each other with respect to a given group of operations if one is obtained from the other by some of the operations (and vice versa).
[edit]Mathematical model for symmetry
The set of all symmetry operations considered, on all objects in a set X, can be modeled as a group action g : G × X → X, where the image of g in G and x in X is written as g·x. If, for some g, g·x = y then x and y are said to be symmetrical to each other. For each object x, operations g for which g·x = x form a group, the symmetry group of the object, a subgroup of G. If the symmetry group of x is the trivial group then x is said to be asymmetric, otherwise symmetric.
A general example is that G is a group of bijections g: V → V acting on the set of functions x: V → W by (gx)(v) = x[g−1(v)] (or a restricted set of such functions that is closed under the group action). Thus a group of bijections of space induces a group action on "objects" in it. The symmetry group of x consists of all g for which x(v) = x[g(v)] for all v. G is the symmetry group of the space itself, and of any object that is uniform throughout space. Some subgroups of G may not be the symmetry group of any object. For example, if the group contains for every v and w in V a g such that g(v) = w, then only the symmetry groups of constant functions x contain that group. However, the symmetry group of constant functions is G itself.
In a modified version for vector fields, we have (gx)(v) = h(g, x[g−1(v)]) where h rotates any vectors and pseudovectors in x, and inverts any vectors (but not pseudovectors) according to rotation and inversion in g, see symmetry in physics. The symmetry group of x consists of all g for which x(v) = h(g, x[g(v)]) for all v. In this case the symmetry group of a constant function may be a proper subgroup of G: a constant vector has only rotational symmetry with respect to rotation about an axis if that axis is in the direction of the vector, and only inversion symmetry if it is zero.
For a common notion of symmetry in Euclidean space, G is the Euclidean group E(n), the group of isometries, and V is the Euclidean space. The rotation group of an object is the symmetry group if G is restricted to E+(n), the group of direct isometries. (For generalizations, see the next subsection.) Objects can be modeled as functions x, of which a value may represent a selection of properties such as color, density, chemical composition, etc. Depending on the selection we consider just symmetries of sets of points (x is just a Boolean function of position v), or, at the other extreme; e.g., symmetry of right and left hand with all their structure.
For a given symmetry group, the properties of part of the object, fully define the whole object. Considering points equivalent which, due to the symmetry, have the same properties, the equivalence classes are the orbits of the group action on the space itself. We need the value of x at one point in every orbit to define the full object. A set of such representatives forms a fundamental domain. The smallest fundamental domain does not have a symmetry; in this sense, one can say that symmetry relies upon asymmetry.
An object with a desired symmetry can be produced by choosing for every orbit a single function value. Starting from a given object x we can, e.g.:
Take the values in a fundamental domain (i.e., add copies of the object).
Take for each orbit some kind of average or sum of the values of x at the points of the orbit (ditto, where the copies may overlap).
If it is desired to have no more symmetry than that in the symmetry group, then the object to be copied should be asymmetric.
As pointed out above, some groups of isometries are not the symmetry group of any object, except in the modified model for vector fields. For example, this applies in 1D for the group of all translations. The fundamental domain is only one point, so we can not make it asymmetric, so any "pattern" invariant under translation is also invariant under reflection (these are the uniform "patterns").
In the vector field version continuous translational symmetry does not imply reflectional symmetry: the function value is constant, but if it contains nonzero vectors, there is no reflectional symmetry. If there is also reflectional symmetry, the constant function value contains no nonzero vectors, but it may contain nonzero pseudovectors. A corresponding 3D example is an infinite cylinder with a current perpendicular to the axis; the magnetic field (a pseudovector) is, in the direction of the cylinder, constant, but nonzero. For vectors (in particular the current density) we have symmetry in every plane perpendicular to the cylinder, as well as cylindrical symmetry. This cylindrical symmetry without mirror planes through the axis is also only possible in the vector field version of the symmetry concept. A similar example is a cylinder rotating about its axis, where magnetic field and current density are replaced by angular momentum and velocity, respectively.
A symmetry group is said to act transitively on a repeated feature of an object if, for every pair of occurrences of the feature there is a symmetry operation mapping the first to the second. For example, in 1D, the symmetry group of {…, 1, 2, 5, 6, 9, 10, 13, 14, …} acts transitively on all these points, while {…, 1, 2, 3, 5, 6, 7, 9, 10, 11, 13, 14, 15, …} does not act transitively on all points. Equivalently, the first set is only one conjugacy class with respect to isometries, while the second has two classes.
[edit]Symmetric functions
Main article: symmetric function
A symmetric function is a function which is unchanged by any permutation of its variables. For example, x + y + z and xy + yz + xz are symmetric functions, whereas x2 – yz is not.
A function may be unchanged by a sub-group of all the permutations of its variables. For example, ac + 3ab + bc is unchanged if a and b are exchanged; its symmetry group is isomorphic to C2.
[edit]Symmetry in logic
A dyadic relation R is symmetric if and only if, whenever it's true that Rab, it's true that Rba. Thus, "is the same age as" is symmetrical, for if Paul is the same age as Mary, then Mary is the same age as Paul.
Symmetric binary logical connectives are and (∧, or &), or (∨, or |), biconditional (if and only if) (↔), nand (not-and, or ⊼), xor (not-biconditional, or ⊻), and nor (not-or, or ⊽).
[edit]In science

[edit]Symmetry in physics
Main article: Symmetry in physics
Symmetry in physics has been generalized to mean invariance—that is, lack of change—under any kind of transformation, for example arbitrary coordinate transformations. This concept has become one of the most powerful tools of theoretical physics, as it has become evident that practically all laws of nature originate in symmetries. In fact, this role inspired the Nobel laureate PW Anderson to write in his widely read 1972 article More is Different that "it is only slightly overstating the case to say that physics is the study of symmetry." See Noether's theorem (which, in greatly simplified form, states that for every continuous mathematical symmetry, there is a corresponding conserved quantity; a conserved current, in Noether's original language); and also, Wigner's classification, which says that the symmetries of the laws of physics determine the properties of the particles found in nature.
[edit]Symmetry in physical objects
[edit]Classical objects
Although an everyday object may appear exactly the same after a symmetry operation such as a rotation or an exchange of two identical parts has been performed on it, it is readily apparent that such a symmetry is true only as an approximation for any ordinary physical object.
For example, if one rotates a precisely machined aluminum equilateral triangle 120 degrees around its center, a casual observer brought in before and after the rotation will be unable to decide whether or not such a rotation took place. However, the reality is that each corner of a triangle will always appear unique when examined with sufficient precision. An observer armed with sufficiently detailed measuring equipment such as optical or electron microscopes will not be fooled; he will immediately recognize that the object has been rotated by looking for details such as crystals or minor deformities.
Such simple thought experiments show that assertions of symmetry in everyday physical objects are always a matter of approximate similarity rather than of precise mathematical sameness. The most important consequence of this approximate nature of symmetries in everyday physical objects is that such symmetries have minimal or no impacts on the physics of such objects. Consequently, only the deeper symmetries of space and time play a major role in classical physics; that is, the physics of large, everyday objects.
[edit]Quantum objects
Remarkably, there exists a realm of physics for which mathematical assertions of simple symmetries in real objects cease to be approximations. That is the domain of quantum physics, which for the most part is the physics of very small, very simple objects such as electrons, protons, light, and atoms.
Unlike everyday objects, objects such as electrons have very limited numbers of configurations, called states, in which they can exist. This means that when symmetry operations such as exchanging the positions of components are applied to them, the resulting new configurations often cannot be distinguished from the originals no matter how diligent an observer is. Consequently, for sufficiently small and simple objects the generic mathematical symmetry assertion F(x) = x ceases to be approximate, and instead becomes an experimentally precise and accurate description of the situation in the real world.
[edit]Consequences of quantum symmetry
While it makes sense that symmetries could become exact when applied to very simple objects, the immediate intuition is that such a detail should not affect the physics of such objects in any significant way. This is in part because it is very difficult to view the concept of exact similarity as physically meaningful. Our mental picture of such situations is invariably the same one we use for large objects: We picture objects or configurations that are very, very similar, but for which if we could "look closer" we would still be able to tell the difference.
However, the assumption that exact symmetries in very small objects should not make any difference in their physics was discovered in the early 1900s to be spectacularly incorrect. The situation was succinctly summarized by Richard Feynman in the direct transcripts of his Feynman Lectures on Physics, Volume III, Section 3.4, Identical particles. (Unfortunately, the quote was edited out of the printed version of the same lecture.)
… if there is a physical situation in which it is impossible to tell which way it happened, it always interferes; it never fails.
The word "interferes" in this context is a quick way of saying that such objects fall under the rules of quantum mechanics, in which they behave more like waves that interfere than like everyday large objects.
In short, when an object becomes so simple that a symmetry assertion of the form F(x) = x becomes an exact statement of experimentally verifiable sameness, x ceases to follow the rules of classical physics and must instead be modeled using the more complex, and often far less intuitive, rules of quantum physics.
This transition also provides an important insight into why the mathematics of symmetry are so deeply intertwined with those of quantum mechanics. When physical systems make the transition from symmetries that are approximate to ones that are exact, the mathematical expressions of those symmetries cease to be approximations and are transformed into precise definitions of the underlying nature of the objects. From that point on, the correlation of such objects to their mathematical descriptions becomes so close that it is difficult to separate the two.
[edit]Generalizations of symmetry
If we have a given set of objects with some structure, then it is possible for a symmetry to merely convert only one object into another, instead of acting upon all possible objects simultaneously. This requires a generalization from the concept of symmetry group to that of a groupoid. Indeed, A. Connes in his book "Non-commutative geometry" writes that Heisenberg discovered quantum mechanics by considering the groupoid of transitions of the hydrogen spectrum.
The notion of groupoid also leads to notions of multiple groupoids, namely sets with many compatible groupoid structures, a structure which trivialises to abelian groups if one restricts to groups. This leads to prospects of higher order symmetry which have been a little explored, as follows.
The automorphisms of a set, or a set with some structure, form a group, which models a homotopy 1-type. The automorphisms of a group G naturally form a crossed module , and crossed modules give an algebraic model of homotopy 2-types. At the next stage, automorphisms of a crossed module fit into a structure known as a crossed square, and this structure is known to give an algebraic model of homotopy 3-types. It is not known how this procedure of generalising symmetry may be continued, although crossed n-cubes have been defined and used in algebraic topology, and these structures are only slowly being brought into theoretical physics.[7][8]
Physicists have come up with other directions of generalization, such as supersymmetry and quantum groups, yet the different options are indistinguishable during various circumstances.
[edit]Symmetry in biology
Further information: symmetry (biology) and facial symmetry
[edit]Symmetry in chemistry
Main article: molecular symmetry
Symmetry is important to chemistry because it explains observations in spectroscopy, quantum chemistry and crystallography. It draws heavily on group theory.
[edit]In history, religion, and culture

In any human endeavor for which an impressive visual result is part of the desired objective, symmetries play a profound role. The innate appeal of symmetry can be found in our reactions to happening across highly symmetrical natural objects, such as precisely formed crystals or beautifully spiraled seashells. Our first reaction in finding such an object often is to wonder whether we have found an object created by a fellow human, followed quickly by surprise that the symmetries that caught our attention are derived from nature itself. In both reactions we give away our inclination to view symmetries both as beautiful and, in some fashion, informative of the world around us.[citation needed]
[edit]Symmetry in social interactions
People observe the symmetrical nature, often including asymmetrical balance, of social interactions in a variety of contexts. These include assessments of reciprocity, empathy, apology, dialog, respect, justice, and revenge. Symmetrical interactions send the message "we are all the same" while asymmetrical interactions send the message "I am special; better than you." Peer relationships are based on symmetry, power relationships are based on asymmetry.[9]
[edit]Symmetry in architecture


The ceiling of Lotfollah mosque, Isfahan, Iran has rotational symmetry of order eight and eight lines of reflection.


Leaning Tower of Pisa


The Taj Mahal has bilateral symmetry.
Another human endeavor in which the visual result plays a vital part in the overall result is architecture. Both in ancient and modern times, the ability of a large structure to impress or even intimidate its viewers has often been a major part of its purpose, and the use of symmetry is an inescapable aspect of how to accomplish such goals.
Just a few examples of ancient architectures that made powerful use of symmetry to impress those around them included the Egyptian Pyramids, the Greek Parthenon, the first and second Temple of Jerusalem, China's Forbidden City, Cambodia's Angkor Wat complex, and the many temples and pyramids of ancient Pre-Columbian civilizations. More recent historical examples of architectures emphasizing symmetries include Gothic architecture cathedrals, and American President Thomas Jefferson's Monticello home. The Taj Mahal is also an example of symmetry.[10]
An interesting example of a broken symmetry in architecture is the Leaning Tower of Pisa, whose notoriety stems in no small part not for the intended symmetry of its design, but for the violation of that symmetry from the lean that developed while it was still under construction. Modern examples of architectures that make impressive or complex use of various symmetries include Australia's Sydney Opera House and Houston, Texas's simpler Astrodome.
Symmetry finds its ways into architecture at every scale, from the overall external views, through the layout of the individual floor plans, and down to the design of individual building elements such as intricately carved doors, stained glass windows, tile mosaics, friezes, stairwells, stair rails, and balustrades. For sheer complexity and sophistication in the exploitation of symmetry as an architectural element, Islamic buildings such as the Taj Mahal often eclipse those of other cultures and ages, due in part to the general prohibition of Islam against using images of people or animals.[11][12]
[edit]Symmetry in pottery and metal vessels


Persian vessel (4th millennium BC)
Since the earliest uses of pottery wheels to help shape clay vessels, pottery has had a strong relationship to symmetry. As a minimum, pottery created using a wheel necessarily begins with full rotational symmetry in its cross-section, while allowing substantial freedom of shape in the vertical direction. Upon this inherently symmetrical starting point cultures from ancient times have tended to add further patterns that tend to exploit or in many cases reduce the original full rotational symmetry to a point where some specific visual objective is achieved. For example, Persian pottery dating from the fourth millennium BC and earlier used symmetric zigzags, squares, cross-hatchings, and repetitions of figures to produce more complex and visually striking overall designs.
Cast metal vessels lacked the inherent rotational symmetry of wheel-made pottery, but otherwise provided a similar opportunity to decorate their surfaces with patterns pleasing to those who used them. The ancient Chinese, for example, used symmetrical patterns in their bronze castings as early as the 17th century BC. Bronze vessels exhibited both a bilateral main motif and a repetitive translated border design.[13][14][15]
[edit]Symmetry in quilts


Kitchen Kaleidoscope Block
As quilts are made from square blocks (usually 9, 16, or 25 pieces to a block) with each smaller piece usually consisting of fabric triangles, the craft lends itself readily to the application of symmetry.[16]
[edit]Symmetry in carpets and rugs


Persian rug.
A long tradition of the use of symmetry in carpet and rug patterns spans a variety of cultures. American Navajo Indians used bold diagonals and rectangular motifs. Many Oriental rugs have intricate reflected centers and borders that translate a pattern. Not surprisingly, rectangular rugs typically use quadrilateral symmetry—that is, motifs that are reflected across both the horizontal and vertical axes.[17][18]
[edit]Symmetry in music


Major and minor triads on the white piano keys are symmetrical to the D. (compare article) (file)
Symmetry is not restricted to the visual arts. Its role in the history of music touches many aspects of the creation and perception of music.
[edit]Musical form
Symmetry has been used as a formal constraint by many composers, such as the arch (swell) form (ABCBA) used by Steve Reich, Béla Bartók, and James Tenney. In classical music, Bach used the symmetry concepts of permutation and invariance.[19]
[edit]Pitch structures
Symmetry is also an important consideration in the formation of scales and chords, traditional or tonal music being made up of non-symmetrical groups of pitches, such as the diatonic scale or the major chord. Symmetrical scales or chords, such as the whole tone scale, augmented chord, or diminished seventh chord (diminished-diminished seventh), are said to lack direction or a sense of forward motion, are ambiguous as to the key or tonal center, and have a less specific diatonic functionality. However, composers such as Alban Berg, Béla Bartók, and George Perle have used axes of symmetry and/or interval cycles in an analogous way to keys or non-tonal tonal centers.
Perle (1992) explains "C–E, D–F♯, [and] Eb–G, are different instances of the same interval … the other kind of identity. … has to do with axes of symmetry. C–E belongs to a family of symmetrically related dyads as follows:"
D D♯ E F F♯ G G♯
D C♯ C B A♯ A G♯
Thus in addition to being part of the interval-4 family, C–E is also a part of the sum-4 family (with C equal to 0).
+ 2 3 4 5 6 7 8
2 1 0 11 10 9 8
4 4 4 4 4 4 4
Interval cycles are symmetrical and thus non-diatonic. However, a seven pitch segment of C5 (the cycle of fifths, which are enharmonic with the cycle of fourths) will produce the diatonic major scale. Cyclic tonal progressions in the works of Romantic composers such as Gustav Mahler and Richard Wagner form a link with the cyclic pitch successions in the atonal music of Modernists such as Bartók, Alexander Scriabin, Edgard Varèse, and the Vienna school. At the same time, these progressions signal the end of tonality.
The first extended composition consistently based on symmetrical pitch relations was probably Alban Berg's Quartet, Op. 3 (1910). (Perle, 1990)
[edit]Equivalency
Tone rows or pitch class sets which are invariant under retrograde are horizontally symmetrical, under inversion vertically. See also Asymmetric rhythm.
[edit]Symmetry in other arts and crafts

Celtic knotwork
The concept of symmetry is applied to the design of objects of all shapes and sizes. Other examples include beadwork, furniture, sand paintings, knotwork, masks, musical instruments, and many other endeavors.
[edit]Symmetry in aesthetics
Main article: Symmetry (physical attractiveness)
The relationship of symmetry to aesthetics is complex. Certain simple symmetries, and in particular bilateral symmetry, seem to be deeply ingrained in the inherent perception by humans of the likely health or fitness of other living creatures, as can be seen by the simple experiment of distorting one side of the image of an attractive face and asking viewers to rate the attractiveness of the resulting image. Consequently, such symmetries that mimic biology tend to have an innate appeal that in turn drives a powerful tendency to create artifacts with similar symmetry. One only needs to imagine the difficulty in trying to market a highly asymmetrical car or truck to general automotive buyers to understand the power of biologically inspired symmetries such as bilateral symmetry.
Another more subtle appeal of symmetry is that of simplicity, which in turn has an implication of safety, security, and familiarity.[citation needed] A highly symmetrical room, for example, is unavoidably also a room in which anything out of place or potentially threatening can be identified easily and quickly.[citation needed] For example, people who have grown up in houses full of exact right angles and precisely identical artifacts can find their first experience in staying in a room with no exact right angles and no exactly identical artifacts to be highly disquieting.[citation needed] Symmetry thus can be a source of comfort not only as an indicator of biological health, but also of a safe and well-understood living environment.
Opposed to this is the tendency for excessive symmetry to be perceived as boring or uninteresting. Humans in particular have a powerful desire to exploit new opportunities or explore new possibilities, and an excessive degree of symmetry can convey a lack of such opportunities.[citation needed] Most people display a preference for figures that have a certain degree of simplicity and symmetry, but enough complexity to make them interesting.[20]
Yet another possibility is that when symmetries become too complex or too challenging, the human mind has a tendency to "tune them out" and perceive them in yet another fashion: as noise that conveys no useful information.[citation needed]
Finally, perceptions and appreciation of symmetries are also dependent on cultural background. The far greater use of complex geometric symmetries in many Islamic cultures, for example, makes it more likely that people from such cultures will appreciate such art forms (or, conversely, to rebel against them).[citation needed]
As in many human endeavors, the result of the confluence of many such factors is that effective use of symmetry in art and architecture is complex, intuitive, and highly dependent on the skills of the individuals who must weave and combine such factors within their own creative work. Along with texture, color, proportion, and other factors, symmetry is a powerful ingredient in any such synthesis; one only need to examine the Taj Mahal to powerful role that symmetry plays in determining the aesthetic appeal of an object.
Modernist architecture rejects symmetry, stating only a bad architect relies on symmetry;[citation needed] instead of symmetrical layout of blocks, masses and structures, Modernist architecture relies on wings and balance of masses. This notion of getting rid of symmetry was first encountered in International style. Some people find asymmetrical layouts of buildings and structures revolutionizing; other find them restless, boring and unnatural.
A few examples of the more explicit use of symmetries in art can be found in the remarkable art of M.C. Escher, the creative design of the mathematical concept of a wallpaper group, and the many applications (both mathematical and real world) of tiling.
[edit]See also

Symmetry in statistics
Skewness, asymmetry of a statistical distribution
Symmetry in games and puzzles
Symmetric games
Sudoku
Symmetry in literature
Palindrome
Moral symmetry
Empathy and Sympathy
Golden Rule
Reciprocity
Reflective equilibrium
Tit for tat
Other
Asymmetric rhythm
Asymmetry
Burnside's lemma
Chirality
M.C. Escher
Even and odd functions
Fixed points of isometry groups in Euclidean space – center of symmetry
Gödel, Escher, Bach
Ignacio Matte Blanco
Semimetric, which is sometimes translated as symmetric in Russian texts.
Spacetime symmetries
Spontaneous symmetry breaking
Symmetric relation
Symmetries of polyiamonds
Symmetries of polyominoes
Symmetry (biology)
Symmetry group
Time symmetry
Wallpaper group
[edit]References

^ Penrose, Roger (2007). Fearful Symmetry. City: Princeton. ISBN 978-0-691-13482-6.
^ a b For example, Aristotle ascribed spherical shape to the heavenly bodies, attributing this formally defined geometric measure of symmetry to the natural order and perfection of the cosmos.
^ Weyl 1982
^ For example, operations such as moving across a regularly patterned tile floor or rotating an eight-sided vase, or complex transformations of equations or in the way music is played.
^ See, e.g., Mainzer, Klaus (2005). Symmetry And Complexity: The Spirit and Beauty of Nonlinear Science. World Scientific. ISBN 981-256-192-7.
^ Symmetric objects can be material, such as a person, crystal, quilt, floor tiles, or molecule, or it can be an abstract structure such as a mathematical equation or a series of tones (music).
^ a b `Higher dimensional group theory'
^ n-category cafe – discussion of n-groups
^ Emotional Competency Entry describing Symmetry
^ Gregory Neil Derry (2002), What Science Is and How It Works, Princeton University Press, p. 269
^ Williams: Symmetry in Architecture
^ Aslaksen: Mathematics in Art and Architecture
^ Chinavoc: The Art of Chinese Bronzes
^ Grant: Iranian Pottery in the Oriental Institute
^ The Metropolitan Museum of Art – Islamic Art
^ Quate: Exploring Geometry Through Quilts
^ Mallet: Tribal Oriental Rugs
^ Dilucchio: Navajo Rugs
^ see ("Fugue No. 21," pdf or Shockwave)
^ Arnheim, Rudolf (1969). Visual Thinking. University of California Press.
[edit]External links

Look up symmetry in Wiktionary, the free dictionary.
Wikimedia Commons has media related to: Symmetry
Calotta: A World of Symmetry
Dutch: Symmetry Around a Point in the Plane
Chapman: Aesthetics of Symmetry
ISIS Symmetry
International Symmetry Association – ISA
Institute Symmetrion
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Supersymmetry
From Wikipedia, the free encyclopedia
(Redirected from Super symmetry)
Beyond the Standard Model

Simulated Large Hadron Collider CMS particle detector data depicting a Higgs boson produced by colliding protons decaying into hadron jets and electrons
Standard Model
Evidence[show]
Theories[show]
Supersymmetry[show]
Quantum gravity[show]
Experiments[show]
v t e
In particle physics, supersymmetry (often abbreviated SUSY) is a symmetry that relates elementary particles of one spin to other particles that differ by half a unit of spin and are known as superpartner. In a theory with unbroken supersymmetry, for every type of boson there exists a corresponding type of fermion with the same mass and internal quantum numbers (other than spin), and vice-versa.
There is no direct evidence for the existence of supersymmetry.[1] It is motivated by possible solutions to several theoretical problems. Since the superpartners of the Standard Model particles have not been observed, supersymmetry must be a broken symmetry if it is a true symmetry of nature. This would allow the superparticles to be heavier than the corresponding Standard Model particles.
If supersymmetry exists close to the TeV energy scale, it allows for a solution of the hierarchy problem of the Standard Model, i.e., the fact that the Higgs boson mass is subject to quantum corrections which — barring extremely fine-tuned cancellations among independent contributions — would make it so large as to undermine the internal consistency of the theory. In supersymmetric theories, on the other hand, the contributions to the quantum corrections coming from Standard Model are naturally canceled by the contributions of the corresponding superpartners. Other attractive features of TeV-scale supersymmetry are the fact that it allows for the high-energy unification of the weak interactions, the strong interactions and electromagnetism, and the fact that it provides a candidate for dark matter and a natural mechanism for electroweak symmetry breaking. Therefore, scenarios where supersymmetric partners appear with masses not much greater than 1 TeV are considered the most well-motivated by theorists.[2] These scenarios would imply that experimental traces of the superpartners should begin to emerge in high-energy collisions at the LHC relatively soon. As of September 2011, no meaningful signs of the superpartners have been observed,[3][4] which is beginning to significantly constrain the most popular incarnations of supersymmetry. However, the total parameter space of consistent supersymmetric extensions of the Standard Model is extremely diverse and can not be definitively ruled out at the LHC.
Another theoretically appealing property of supersymmetry is that it offers the only "loophole" to the Coleman–Mandula theorem, which prohibits spacetime and internal symmetries from being combined in any nontrivial way, for quantum field theories like the Standard Model under very general assumptions. The Haag-Lopuszanski-Sohnius theorem demonstrates that supersymmetry is the only way spacetime and internal symmetries can be consistently combined.[5]
In general, supersymmetric quantum field theory is often much easier to work with, as many more problems become exactly solvable. Supersymmetry is also a feature of most versions of string theory, though it may exist in nature even if string theories are not experimentally detected.
The Minimal Supersymmetric Standard Model is one of the best studied candidates for physics beyond the Standard Model. Theories of gravity that are also invariant under supersymmetry are known as supergravity theories.
Contents [hide]
1 History
2 Applications
2.1 Extension of possible symmetry groups
2.1.1 The supersymmetry algebra
2.2 The Supersymmetric Standard Model
2.2.1 Gauge Coupling Unification
2.3 Supersymmetric quantum mechanics
2.4 Mathematics
3 General supersymmetry
3.1 Extended supersymmetry
3.2 Supersymmetry in alternate numbers of dimensions
4 Supersymmetry as a quantum group
5 Supersymmetry in quantum gravity
6 Current limits
7 See also
8 References
9 Further reading
10 External links
[edit]History

A supersymmetry relating mesons and baryons was first proposed, in the context of hadronic physics, by Hironari Miyazawa in 1966, but his work was ignored at the time.[6][7][8][9] In the early 1970s, J. L. Gervais and B. Sakita (in 1971), Yu. A. Golfand and E. P. Likhtman (also in 1971), D.V. Volkov and V.P. Akulov (in 1972) and J. Wess and B. Zumino (in 1974) independently rediscovered supersymmetry, a radically new type of symmetry of spacetime and fundamental fields, which establishes a relationship between elementary particles of different quantum nature, bosons and fermions, and unifies spacetime and internal symmetries of the microscopic world. Supersymmetry first[clarification needed] arose in 1971 in the context of an early version of string theory by Pierre Ramond, John H. Schwarz and André Neveu, but the mathematical structure of supersymmetry has subsequently been applied successfully to other areas of physics; firstly by Wess, Zumino, and Abdus Salam and their fellow researchers to particle physics, and later to a variety of fields, ranging from quantum mechanics to statistical physics. It remains a vital part of many proposed theories of physics.
The first realistic supersymmetric version of the Standard Model was proposed in 1981 by Howard Georgi and Savas Dimopoulos and is called the Minimal Supersymmetric Standard Model or MSSM for short. It was proposed to solve the hierarchy problem and predicts superpartners with masses between 100 GeV and 1 TeV. As of 2009 there is no irrefutable experimental evidence that supersymmetry is a symmetry of nature[citation needed]. Since 2010, the Large Hadron Collider at CERN is producing the world's highest energy collisions and offers the best chance at discovering superparticles for the foreseeable future.
[edit]Applications

[edit]Extension of possible symmetry groups
One reason that physicists explored supersymmetry is because it offers an extension to the more familiar symmetries of quantum field theory. These symmetries are grouped into the Poincaré group and internal symmetries and the Coleman–Mandula theorem showed that under certain assumptions, the symmetries of the S-matrix must be a direct product of the Poincaré group with a compact internal symmetry group or if there is no mass gap, the conformal group with a compact internal symmetry group. In 1971 Golfand and Likhtman were the first to show that the Poincaré algebra can be extended through introduction of four anticommuting spinor generators (in four dimensions), which later became known as supercharges. In 1975 the Haag-Lopuszanski-Sohnius theorem analyzed all possible superalgebras in the general form, including those with an extended number of the supergenerators and central charges. This extended super-Poincaré algebra paved the way for obtaining a very large and important class of supersymmetric field theories.
[edit]The supersymmetry algebra
Main article: Supersymmetry algebra
Traditional symmetries in physics are generated by objects that transform under the tensor representations of the Poincaré group and internal symmetries. Supersymmetries, on the other hand, are generated by objects that transform under the spinor representations. According to the spin-statistics theorem, bosonic fields commute while fermionic fields anticommute. Combining the two kinds of fields into a single algebra requires the introduction of a Z2-grading under which the bosons are the even elements and the fermions are the odd elements. Such an algebra is called a Lie superalgebra.
The simplest supersymmetric extension of the Poincaré algebra is the Super-Poincaré algebra. Expressed in terms of two Weyl spinors, has the following anti-commutation relation:

and all other anti-commutation relations between the Qs and commutation relations between the Qs and Ps vanish. In the above expression are the generators of translation and are the Pauli matrices.
There are representations of a Lie superalgebra that are analogous to representations of a Lie algebra. Each Lie algebra has an associated Lie group and a Lie superalgebra can sometimes be extended into representations of a Lie supergroup.
[edit]The Supersymmetric Standard Model
Main article: Minimal Supersymmetric Standard Model
Incorporating supersymmetry into the Standard Model requires doubling the number of particles since there is no way that any of the particles in the Standard Model can be superpartners of each other. With the addition of new particles, there are many possible new interactions. The simplest possible supersymmetric model consistent with the Standard Model is the Minimal Supersymmetric Standard Model (MSSM) which can include the necessary additional new particles that are able to be superpartners of those in the Standard Model.


Cancellation of the Higgs boson quadratic mass renormalization between fermionic top quark loop and scalar stop squark tadpole Feynman diagrams in a supersymmetric extension of the Standard Model
One of the main motivations for SUSY comes from the quadratically divergent contributions to the Higgs mass squared. The quantum mechanical interactions of the Higgs boson causes a large renormalization of the Higgs mass and unless there is an accidental cancellation, the natural size of the Higgs mass is the highest scale possible. This problem is known as the hierarchy problem. Supersymmetry reduces the size of the quantum corrections by having automatic cancellations between fermionic and bosonic Higgs interactions. If supersymmetry is restored at the weak scale, then the Higgs mass is related to supersymmetry breaking which can be induced from small non-perturbative effects explaining the vastly different scales in the weak interactions and gravitational interactions.
In many supersymmetric Standard Models there is a heavy stable particle (such as neutralino) which could serve as a Weakly interacting massive particle (WIMP) dark matter candidate. The existence of a supersymmetric dark matter candidate is closely tied to R-parity.
The standard paradigm for incorporating supersymmetry into a realistic theory is to have the underlying dynamics of the theory be supersymmetric, but the ground state of the theory does not respect the symmetry and supersymmetry is broken spontaneously. The supersymmetry break can not be done permanently by the particles of the MSSM as they currently appear. This means that there is a new sector of the theory that is responsible for the breaking. The only constraint on this new sector is that it must break supersymmetry permanently and must give superparticles TeV scale masses. There are many models that can do this and most of their details do not currently matter. In order to parameterize the relevant features of supersymmetry breaking, arbitrary soft SUSY breaking terms are added to the theory which temporarily break SUSY explicitly but could never arise from a complete theory of supersymmetry breaking.
[edit]Gauge Coupling Unification
Main article: Minimal_Supersymmetric_Standard_Model#Gauge_Coupling_Unification
One piece of evidence for supersymmetry existing is gauge coupling unification. The renormalization group evolution of the three gauge coupling constants of the Standard Model is somewhat sensitive to the present particle content of the theory. These coupling constants do not quite meet together at a common energy scale if we run the renormalization group using the Standard Model.[1] With the addition of minimal SUSY joint convergence of the coupling constants is projected at approximately 1016 GeV.[1]
[edit]Supersymmetric quantum mechanics
Main article: Supersymmetric quantum mechanics
Supersymmetric quantum mechanics adds the SUSY superalgebra to quantum mechanics as opposed to quantum field theory. Supersymmetric quantum mechanics often comes up when studying the dynamics of supersymmetric solitons and due to the simplified nature of having fields only functions of time (rather than space-time), a great deal of progress has been made in this subject and is now studied in its own right.
SUSY quantum mechanics involves pairs of Hamiltonians which share a particular mathematical relationship, which are called partner Hamiltonians. (The potential energy terms which occur in the Hamiltonians are then called partner potentials.) An introductory theorem shows that for every eigenstate of one Hamiltonian, its partner Hamiltonian has a corresponding eigenstate with the same energy. This fact can be exploited to deduce many properties of the eigenstate spectrum. It is analogous to the original description of SUSY, which referred to bosons and fermions. We can imagine a "bosonic Hamiltonian", whose eigenstates are the various bosons of our theory. The SUSY partner of this Hamiltonian would be "fermionic", and its eigenstates would be the theory's fermions. Each boson would have a fermionic partner of equal energy.
SUSY concepts have provided useful extensions to the WKB approximation. In addition, SUSY has been applied to non-quantum statistical mechanics through the Fokker-Planck equation.
[edit]Mathematics
SUSY is also sometimes studied mathematically for its intrinsic properties. This is because it describes complex fields satisfying a property known as holomorphy, which allows holomorphic quantities to be exactly computed. This makes supersymmetric models useful toy models of more realistic theories. A prime example of this has been the demonstration of S-duality in four-dimensional gauge theories that interchanges particles and monopoles.
[edit]General supersymmetry

Supersymmetry appears in many different contexts in theoretical physics that are closely related. It is possible to have multiple supersymmetries and also have supersymmetric extra dimensions.
[edit]Extended supersymmetry
Main article: Extended supersymmetry
It is possible to have more than one kind of supersymmetry transformation. Theories with more than one supersymmetry transformation are known as extended supersymmetric theories. The more supersymmetry a theory has, the more constrained the field content and interactions are. Typically the number of copies of a supersymmetry is a power of 2, i.e. 1, 2, 4, 8. In four dimensions, a spinor has four degrees of freedom and thus the minimal number of supersymmetry generators is four in four dimensions and having eight copies of supersymmetry means that there are 32 supersymmetry generators.
The maximal number of supersymmetry generators possible is 32. Theories with more than 32 supersymmetry generators automatically have massless fields with spin greater than 2. It is not known how to make massless fields with spin greater than two interact, so the maximal number of supersymmetry generators considered is 32. This corresponds to an N = 8 supersymmetry theory. Theories with 32 supersymmetries automatically have a graviton.
In four dimensions there are the following theories, with the corresponding multiplets[10](CPT adds a copy, whenever they are not invariant under such symmetry)
N = 1
Chiral multiplet: (0,1⁄2) Vector multiplet: (1⁄2,1) Gravitino multiplet: (1,3⁄2) Graviton multiplet: (3⁄2,2)
N = 2
hypermultiplet: (-1⁄2,02,1⁄2) vector multiplet: (0,1⁄22,1) supergravity multiplet: (1,3⁄22,2)
N = 4
Vector multiplet: (-1,-1⁄24,06,1⁄24,1) Supergravity multiplet: (0,1⁄24,16,3⁄24,2)
N = 8
Supergravity multiplet: (-2,-3⁄28,-128,-1⁄256,070,1⁄256,128,3⁄28,2)
[edit]Supersymmetry in alternate numbers of dimensions
It is possible to have supersymmetry in dimensions other than four. Because the properties of spinors change drastically between different dimensions, each dimension has its characteristic. In d dimensions, the size of spinors is roughly 2d/2 or 2(d − 1)/2. Since the maximum number of supersymmetries is 32, the greatest number of dimensions in which a supersymmetric theory can exist is eleven.
[edit]Supersymmetry as a quantum group

Main article: Supersymmetry as a quantum group
Supersymmetry can be reinterpreted in the language of noncommutative geometry and quantum groups. In particular, it involves a mild form of noncommutativity, namely supercommutativity. See the main article for more details.
[edit]Supersymmetry in quantum gravity

Supersymmetry is part of a larger enterprise of theoretical physics to unify everything we know about the physical world into a single fundamental framework of physical laws, known as the quest for a Theory of Everything (TOE). A significant part of this larger enterprise is the quest for a theory of quantum gravity, which would unify the classical theory of general relativity and the Standard Model, which explains the other three basic forces in physics (electromagnetism, the strong interaction, and the weak interaction), and provides a palette of fundamental particles upon which all four forces act. Two of the most active approaches to forming a theory of quantum gravity are string theory and loop quantum gravity (LQG), although in theory, supersymmetry could be a component of other theoretical approaches as well.
For string theory to be consistent, supersymmetry appears to be required at some level (although it may be a strongly broken symmetry). In particle theory, supersymmetry is recognized as a way to stabilize the hierarchy between the unification scale and the electroweak scale (or the Higgs boson mass), and can also provide a natural dark matter candidate. String theory also requires extra spatial dimensions which have to be compactified as in Kaluza-Klein theory.
Loop quantum gravity (LQG), in its current formulation, predicts no additional spatial dimensions, nor anything else about particle physics. These theories can be formulated in three spatial dimensions and one dimension of time, although in some LQG theories dimensionality is an emergent property of the theory, rather than a fundamental assumption of the theory. Also, LQG is a theory of quantum gravity which does not require supersymmetry. Lee Smolin, one of the originators of LQG, has proposed that a loop quantum gravity theory incorporating either supersymmetry or extra dimensions, or both, be called "loop quantum gravity II".
If experimental evidence confirms supersymmetry in the form of supersymmetric particles such as the neutralino that is often believed to be the lightest superpartner, some people believe this would be a major boost to string theory. Since supersymmetry is a required component of string theory, any discovered supersymmetry would be consistent with string theory. If the Large Hadron Collider and other major particle physics experiments fail to detect supersymmetric partners or evidence of extra dimensions, many versions of string theory which had predicted certain low mass superpartners to existing particles may need to be significantly revised. The failure of experiments to discover either supersymmetric partners or extra spatial dimensions, as of 2009, has encouraged loop quantum gravity researchers.
[edit]Current limits

Supersymmetric models are constrained by a variety of experiments, including measurements of low-energy observables, for example the anomalous magnetic moment of the muon at Brookhaven; the WMAP dark matter density measurement and direct detection experiments, for example XENON-100; and by particle collider experiments, including B-physics, Higgs phenomenology and direct searches for sparticles, at the Large Electron–Positron Collider, Tevatron and the LHC.
Historically, the tightest limits were from direct production at colliders. The first mass limits for squarks and gluinos were made at CERN by the UA1 experiment and the UA2 experiment at the Super Proton Synchrotron. LEP later set very strong limits.[11] In 2006 these limits were extended by the D0 experiment[12][13]
From 2003, WMAP's dark matter density measurements have strongly constrained supersymmetry models, which have to be tuned to invoke a particular mechansim to sufficiently reduce the neutralino density.
Prior to the launch of the LHC, in 2009, fits of available data to CMSSM and NUHM1 indicated that squarks and gluinos were most likely to have masses in 500 to 800 GeV range, though values as high as 2.5 TeV were allowed with low probabilities. Neutralinos and sleptons were expected to be quite light, with the lightest neutralino and the lightest stau most likely to be found between 100 to 150 GeV.[14]
As of 2012, the LHC has found no evidence for supersymmetry, and, as a result, has surpassed existing experimental limits from Large Electron–Positron Collider and Tevatron and partially excluded the aforementioned expected ranges.[15][16][17][18] Based on the data sample collected by the CMS detector at the LHC through the summer of 2011, CMSSM squarks have been excluded up to the mass of 1.1 TeV and gluinos have been excluded up to 500 GeV.[19]Searches are only applicable for a finite set of tested points because simulation using the Monte Carlo method must be made so that limits for that particular model can be calculated. This complicates matters because different experiments have looked at different sets of points. Some extrapolation between points can be made within particular models but it is difficult to set general limits even for the Minimal Supersymmetric Standard Model.
In 2011 and 2012, the LHC discovered a Higgs boson boson with a mass of about 125 GeV, and with couplings to fermions and bosons which are consistent with the Standard Model. The MSSM predicts that the mass of the lightest Higgs boson should not be much higher than the mass of the Z boson, and, in the absence of fine tuning (with the supersymmetry breaking scale on the order of 1 TeV), should not exceed 130 GeV. Furthermore, for values of the MSSM parameter tan β ≤ 3, it predicts Higgs mass below 114 GeV over most of the parameter space.[20] This region of Higgs mass was excluded by LEP by 2000. The LHC result is somewhat problematic for the minimal supersymmetric model, as the value of 125 GeV is relatively large for the model and can only be achieved with large radiative loop corrections from top squarks, which many theorists consider to be "unnatural" (see naturalness and fine tuning).[21] Furthermore, in 2012, the LHC measured deviations from Standard Model predicted Higgs couplings, particularly in their gamma-gamma final state, which, if they persist, could severely constrain the MSSM.
In spite of the null searches and the heavy Higgs, a recent analysis of the most constrained supersymmetry model, the CMSSM, suggests that the model is still compatible with all present experimental constraints.[22] The preferred masses for squarks and gluinos is about 2 TeV. The resulting fine-tuning of the Higgs boson mass (see little hierarchy problem) and Z-boson mass (see mu problem), however, is considered "unnatural," and some theorists now favor extended supersymmetry models, for example, the NMSSM.
[edit]See also

Wess–Zumino model
Minimal Supersymmetric Standard Model
Supersymmetry as a quantum group
Quantum group
Supercharge
Superfield
Supergeometry
Supergravity
Supergroup
Superspace
[edit]References

^ a b c Gordon L. Kane, The Dawn of Physics Beyond the Standard Model, Scientific American, June 2003, page 60 and The frontiers of physics, special edition, Vol 15, #3, page 8 "Indirect evidence for supersymmetry comes from the extrapolation of interactions to high energies."
^ http://profmattstrassler.com/articles-and-posts/lhcposts/what-do-current-mid-august-2011-lhc-results-imply-about-supersymmetry/
^ ATLAS SUSY search documents
^ CMS SUSY search documents
^ R. Haag, J. T. Lopuszanski and M. Sohnius, "All Possible Generators Of Supersymmetries Of The S Matrix", Nucl. Phys. B 88 (1975) 257
^ H. Miyazawa (1966). "Baryon Number Changing Currents". Prog. Theor. Phys. 36 (6): 1266–1276. Bibcode 1966PThPh..36.1266M. doi:10.1143/PTP.36.1266.
^ H. Miyazawa (1968). "Spinor Currents and Symmetries of Baryons and Mesons". Phys. Rev. 170 (5): 1586–1590. Bibcode 1968PhRv..170.1586M. doi:10.1103/PhysRev.170.1586.
^ Michio Kaku, Quantum Field Theory, ISBN 0-19-509158-2, pg 663.
^ Peter Freund, Introduction to Supersymmetry, ISBN 0-521-35675-X, pages 26-27, 138.
^ Polchinski,J. String theory. Vol. 2: Superstring theory and beyond, Appendix B
^ LEPSUSYWG, ALEPH, DELPHI, L3 and OPAL experiments, charginos, large m0 LEPSUSYWG/01-03.1
^ The D0-Collaboration, Search for associated production of charginos and neutralinos in the trilepton final state using 2.3 $fb^-1$ of data, arXiv:/0901.0646 [hep-ex]
^ The D0 Collaboration, V. Abazov, et al., Search for Squarks and Gluinos in events with jets and missing transverse energy using 2.1 $fb^-1$ of $p\bar{p}$ collision data at $\sqrt{s}$ = 1.96 TeV , arXiv:0712.3805v2 [hep-ex]
^ O. Buchmueller et al.. Likelihood Functions for Supersymmetric Observables in Frequentist Analyses of the CMSSM and NUHM1.
^ Implications of Initial LHC Searches for Supersymmetry
^ Fine-tuning implications for complementary dark matter and LHC SUSY searches
^ What LHC tells about SUSY
^ Early SUSY searches at the LHC
^ CMS Collaboration; Khachatryan, V.; Sirunyan, A.; Tumasyan, A.; Adam, W.; Bergauer, T.; Dragicevic, M.; Erö, J. et al. (November 2011). "Search for Supersymmetry at the LHC in Events with Jets and Missing Transverse Energy". Physical Review Letters 107 (22). doi:10.1103/PhysRevLett.107.221804.
^ Marcela Carena and Howard E. Haber; Haber (1970). "Higgs Boson Theory and Phenomenology". Progress in Particle and Nuclear Physics 50: 63. arXiv:hep-ph/0208209v3.pdf. Bibcode 2003PrPNP..50...63C. doi:10.1016/S0146-6410(02)00177-1.
^ Patrick Draper et al (December 2011). Implications of a 125 GeV Higgs for the MSSM and Low-Scale SUSY Breaking. arXiv:1112.3068v1.pdf.
^ [[1]]"Global Fits of the cMSSM and NUHM including the LHC Higgs discovery and new XENON100 constraints", C. Strege, G. Bertone, F. Feroz, M. Fornasa, R. Ruiz de Austri, R. Trotta, arXiv:1212.2636
[edit]Further reading

A Supersymmetry Primer by S. Martin, 2011
Introduction to Supersymmetry By Joseph D. Lykken, 1996
An Introduction to Supersymmetry By Manuel Drees, 1996
Introduction to Supersymmetry By Adel Bilal, 2001
An Introduction to Global Supersymmetry by Philip Arygres, 2001
Weak Scale Supersymmetry by Howard Baer and Xerxes Tata, 2006.
Cooper, F., A. Khare and U. Sukhatme. "Supersymmetry in Quantum Mechanics." Phys. Rep. 251 (1995) 267-85 (arXiv:hep-th/9405029).
Junker, G. Supersymmetric Methods in Quantum and Statistical Physics, Springer-Verlag (1996).
Gordon L. Kane.Supersymmetry: Unveiling the Ultimate Laws of Nature Basic Books, New York (2001). ISBN 0-7382-0489-7.
Gordon L. Kane and Shifman, M., eds. The Supersymmetric World: The Beginnings of the Theory, World Scientific, Singapore (2000). ISBN 981-02-4522-X.
D.V. Volkov, V.P. Akulov, Pisma Zh.Eksp.Teor.Fiz. 16 (1972) 621; Phys.Lett. B46 (1973) 109.
V.P. Akulov, D.V. Volkov, Teor.Mat.Fiz. 18 (1974) 39.
Weinberg, Steven, The Quantum Theory of Fields, Volume 3: Supersymmetry, Cambridge University Press, Cambridge, (1999). ISBN 0-521-66000-9.
Wess, Julius, and Jonathan Bagger, Supersymmetry and Supergravity, Princeton University Press, Princeton, (1992). ISBN 0-691-02530-4.
Bennett GW, et al.; Muon (g−2) Collaboration (2004). "Measurement of the negative muon anomalous magnetic moment to 0.7 ppm". Physical Review Letters 92 (16): 161802. arXiv:hep-ex/0401008. Bibcode 2004PhRvL..92p1802B. doi:10.1103/PhysRevLett.92.161802. PMID 15169217.
Brookhaven National Laboratory (Jan. 8, 2004). New g−2 measurement deviates further from Standard Model. Press Release.
Fermi National Accelerator Laboratory (Sept 25, 2006). Fermilab's CDF scientists have discovered the quick-change behavior of the B-sub-s meson. Press Release.
[edit]External links

What do current LHC results (mid-August 2011) imply about supersymmetry? Matt Strassler
ATLAS Experiment Supersymmetry search documents
CMS Experiment Supersymmetry search documents
"Particle wobble shakes up supersymmetry", Cosmos magazine, September 2006
LHC results put supersymmetry theory 'on the spot' BBC news 27/8/2011
SUSY running out of hiding places BBC news 12/11/2012
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Geometry
From Wikipedia, the free encyclopedia
For other uses, see Geometry (disambiguation).


An illustration of Desargues' theorem, an important result in Euclidean and projective geometry.
Geometry

Oxyrhynchus papyrus (P.Oxy. I 29) showing fragment of Euclid's Elements
History of geometry
Branches[show]
Research areas[show]
Important concepts[show]
Geometers[show]
v t e
Geometry (Ancient Greek: γεωμετρία; geo- "earth", -metron "measurement") is a branch of mathematics concerned with questions of shape, size, relative position of figures, and the properties of space. A mathematician who works in the field of geometry is called a geometer. Geometry arose independently in a number of early cultures as a body of practical knowledge concerning lengths, areas, and volumes, with elements of a formal mathematical science emerging in the West as early as Thales (6th Century BC). By the 3rd century BC geometry was put into an axiomatic form by Euclid, whose treatment—Euclidean geometry—set a standard for many centuries to follow.[1] Archimedes developed ingenious techniques for calculating areas and volumes, in many ways anticipating modern integral calculus. The field of astronomy, especially mapping the positions of the stars and planets on the celestial sphere and describing the relationship between movements of celestial bodies, served as an important source of geometric problems during the next one and a half millennia. Both geometry and astronomy were considered in the classical world to be part of the Quadrivium, a subset of the seven liberal arts considered essential for a free citizen to master.
The introduction of coordinates by René Descartes and the concurrent developments of algebra marked a new stage for geometry, since geometric figures, such as plane curves, could now be represented analytically, i.e., with functions and equations. This played a key role in the emergence of infinitesimal calculus in the 17th century. Furthermore, the theory of perspective showed that there is more to geometry than just the metric properties of figures: perspective is the origin of projective geometry. The subject of geometry was further enriched by the study of intrinsic structure of geometric objects that originated with Euler and Gauss and led to the creation of topology and differential geometry.
In Euclid's time there was no clear distinction between physical space and geometrical space. Since the 19th-century discovery of non-Euclidean geometry, the concept of space has undergone a radical transformation, and the question arose: which geometrical space best fits physical space? With the rise of formal mathematics in the 20th century, also 'space' (and 'point', 'line', 'plane') lost its intuitive contents, so today we have to distinguish between physical space, geometrical spaces (in which 'space', 'point' etc. still have their intuitive meaning) and abstract spaces. Contemporary geometry considers manifolds, spaces that are considerably more abstract than the familiar Euclidean space, which they only approximately resemble at small scales. These spaces may be endowed with additional structure, allowing one to speak about length. Modern geometry has multiple strong bonds with physics, exemplified by the ties between pseudo-Riemannian geometry and general relativity. One of the youngest physical theories, string theory, is also very geometric in flavour.
While the visual nature of geometry makes it initially more accessible than other parts of mathematics, such as algebra or number theory, geometric language is also used in contexts far removed from its traditional, Euclidean provenance (for example, in fractal geometry and algebraic geometry).[2]
Contents [hide]
1 Overview
1.1 Practical geometry
1.2 Axiomatic geometry
1.3 Geometric constructions
1.4 Numbers in geometry
1.5 Geometry of position
1.6 Geometry beyond Euclid
1.7 Dimension
1.8 Symmetry
2 History of geometry
3 Contemporary geometry
3.1 Euclidean geometry
3.2 Differential geometry
3.3 Topology and geometry
3.4 Algebraic geometry
4 See also
4.1 Lists
4.2 Related topics
5 References
5.1 Notes
6 Sources
6.1 Bibliography
7 External links
[edit]Overview



Visual proof of the Pythagorean theorem for the (3, 4, 5) triangle as in the Chou Pei Suan Ching 500–200 BC.
The recorded development of geometry spans more than two millennia. It is hardly surprising that perceptions of what constituted geometry evolved throughout the ages.
[edit]Practical geometry
Geometry originated as a practical science concerned with surveying, measurements, areas, and volumes. Among the notable accomplishments one finds formulas for lengths, areas and volumes, such as Pythagorean theorem, circumference and area of a circle, area of a triangle, volume of a cylinder, sphere, and a pyramid. A method of computing certain inaccessible distances or heights based on similarity of geometric figures is attributed to Thales. Development of astronomy led to emergence of trigonometry and spherical trigonometry, together with the attendant computational techniques.
[edit]Axiomatic geometry


An illustration of Euclid's parallel postulate
See also: Euclidean geometry
Euclid took a more abstract approach in his Elements, one of the most influential books ever written. Euclid introduced certain axioms, or postulates, expressing primary or self-evident properties of points, lines, and planes. He proceeded to rigorously deduce other properties by mathematical reasoning. The characteristic feature of Euclid's approach to geometry was its rigor, and it has come to be known as axiomatic or synthetic geometry. At the start of the 19th century the discovery of non-Euclidean geometries by Gauss, Lobachevsky, Bolyai, and others led to a revival of interest, and in the 20th century David Hilbert employed axiomatic reasoning in an attempt to provide a modern foundation of geometry.


Geometry lessons in the 20th century
[edit]Geometric constructions
Main article: Compass and straightedge constructions
Classical geometers paid special attention to constructing geometric objects that had been described in some other way. Classically, the only instruments allowed in geometric constructions are the compass and straightedge. Also, every construction had to be complete in a finite number of steps. However, some problems turned out to be difficult or impossible to solve by these means alone, and ingenious constructions using parabolas and other curves, as well as mechanical devices, were found.
[edit]Numbers in geometry


The Pythagoreans discovered that the sides of a triangle could have incommensurable lengths.
In ancient Greece the Pythagoreans considered the role of numbers in geometry. However, the discovery of incommensurable lengths, which contradicted their philosophical views, made them abandon abstract numbers in favor of concrete geometric quantities, such as length and area of figures. Numbers were reintroduced into geometry in the form of coordinates by Descartes, who realized that the study of geometric shapes can be facilitated by their algebraic representation, and whom the Cartesian plane is named after. Analytic geometry applies methods of algebra to geometric questions, typically by relating geometric curves and algebraic equations. These ideas played a key role in the development of calculus in the 17th century and led to discovery of many new properties of plane curves. Modern algebraic geometry considers similar questions on a vastly more abstract level.
[edit]Geometry of position
Main articles: Projective geometry and Topology
Even in ancient times, geometers considered questions of relative position or spatial relationship of geometric figures and shapes. Some examples are given by inscribed and circumscribed circles of polygons, lines intersecting and tangent to conic sections, the Pappus and Menelaus configurations of points and lines. In the Middle Ages new and more complicated questions of this type were considered: What is the maximum number of spheres simultaneously touching a given sphere of the same radius (kissing number problem)? What is the densest packing of spheres of equal size in space (Kepler conjecture)? Most of these questions involved 'rigid' geometrical shapes, such as lines or spheres. Projective, convex and discrete geometry are three sub-disciplines within present day geometry that deal with these and related questions.
Leonhard Euler, in studying problems like the Seven Bridges of Königsberg, considered the most fundamental properties of geometric figures based solely on shape, independent of their metric properties. Euler called this new branch of geometry geometria situs (geometry of place), but it is now known as topology. Topology grew out of geometry, but turned into a large independent discipline. It does not differentiate between objects that can be continuously deformed into each other. The objects may nevertheless retain some geometry, as in the case of hyperbolic knots.
[edit]Geometry beyond Euclid


Differential geometry uses tools from calculus to study problems in geometry.
For nearly two thousand years since Euclid, while the range of geometrical questions asked and answered inevitably expanded, basic understanding of space remained essentially the same. Immanuel Kant argued that there is only one, absolute, geometry, which is known to be true a priori by an inner faculty of mind: Euclidean geometry was synthetic a priori.[3] This dominant view was overturned by the revolutionary discovery of non-Euclidean geometry in the works of Gauss (who never published his theory), Bolyai, and Lobachevsky, who demonstrated that ordinary Euclidean space is only one possibility for development of geometry. A broad vision of the subject of geometry was then expressed by Riemann in his 1867 inauguration lecture Über die Hypothesen, welche der Geometrie zu Grunde liegen (On the hypotheses on which geometry is based),[4] published only after his death. Riemann's new idea of space proved crucial in Einstein's general relativity theory and Riemannian geometry, which considers very general spaces in which the notion of length is defined, is a mainstay of modern geometry.
[edit]Dimension
Where the traditional geometry allowed dimensions 1 (a line), 2 (a plane) and 3 (our ambient world conceived of as three-dimensional space), mathematicians have used higher dimensions for nearly two centuries. Dimension has gone through stages of being any natural number n, possibly infinite with the introduction of Hilbert space, and any positive real number in fractal geometry. Dimension theory is a technical area, initially within general topology, that discusses definitions; in common with most mathematical ideas, dimension is now defined rather than an intuition. Connected topological manifolds have a well-defined dimension; this is a theorem (invariance of domain) rather than anything a priori.
The issue of dimension still matters to geometry, in the absence of complete answers to classic questions. Dimensions 3 of space and 4 of space-time are special cases in geometric topology. Dimension 10 or 11 is a key number in string theory. Research may bring a satisfactory geometric reason for the significance of 10 and 11 dimensions.
[edit]Symmetry


A tiling of the hyperbolic plane
The theme of symmetry in geometry is nearly as old as the science of geometry itself. Symmetric shapes such as the circle, regular polygons and platonic solids held deep significance for many ancient philosophers and were investigated in detail before the time of Euclid. Symmetric patterns occur in nature and were artistically rendered in a multitude of forms, including the graphics of M. C. Escher. Nonetheless, it was not until the second half of 19th century that the unifying role of symmetry in foundations of geometry was recognized. Felix Klein's Erlangen program proclaimed that, in a very precise sense, symmetry, expressed via the notion of a transformation group, determines what geometry is. Symmetry in classical Euclidean geometry is represented by congruences and rigid motions, whereas in projective geometry an analogous role is played by collineations, geometric transformations that take straight lines into straight lines. However it was in the new geometries of Bolyai and Lobachevsky, Riemann, Clifford and Klein, and Sophus Lie that Klein's idea to 'define a geometry via its symmetry group' proved most influential. Both discrete and continuous symmetries play prominent roles in geometry, the former in topology and geometric group theory, the latter in Lie theory and Riemannian geometry.
A different type of symmetry is the principle of duality in projective geometry (see Duality (projective geometry)) among other fields. This meta-phenomenon can roughly be described as follows: in any theorem, exchange point with plane, join with meet, lies in with contains, and you will get an equally true theorem. A similar and closely related form of duality exists between a vector space and its dual space.
[edit]History of geometry

Main article: History of geometry


A European and an Arab practicing geometry in the 15th century.
The earliest recorded beginnings of geometry can be traced to ancient Mesopotamia and Egypt in the 2nd millennium BC.[5][6] Early geometry was a collection of empirically discovered principles concerning lengths, angles, areas, and volumes, which were developed to meet some practical need in surveying, construction, astronomy, and various crafts. The earliest known texts on geometry are the Egyptian Rhind Papyrus (2000–1800 BC) and Moscow Papyrus (c. 1890 BC), the Babylonian clay tablets such as Plimpton 322 (1900 BC). For example, the Moscow Papyrus gives a formula for calculating the volume of a truncated pyramid, or frustum.[7] South of Egypt the ancient Nubians established a system of geometry including early versions of sun clocks.[8][9]
In the 7th century BC, the Greek mathematician Thales of Miletus used geometry to solve problems such as calculating the height of pyramids and the distance of ships from the shore. He is credited with the first use of deductive reasoning applied to geometry, by deriving four corollaries to Thales' Theorem.[10] Pythagoras established the Pythagorean School, which is credited with the first proof of the Pythagorean theorem,[11] though the statement of the theorem has a long history[12][13] Eudoxus (408–c.355 BC) developed the method of exhaustion, which allowed the calculation of areas and volumes of curvilinear figures,[14] as well as a theory of ratios that avoided the problem of incommensurable magnitudes, which enabled subsequent geometers to make significant advances. Around 300 BC, geometry was revolutionized by Euclid, whose Elements, widely considered the most successful and influential textbook of all time,[15] introduced mathematical rigor through the axiomatic method and is the earliest example of the format still used in mathematics today, that of definition, axiom, theorem, and proof. Although most of the contents of the Elements were already known, Euclid arranged them into a single, coherent logical framework.[16] The Elements was known to all educated people in the West until the middle of the 20th century and its contents are still taught in geometry classes today.[17] Archimedes (c.287–212 BC) of Syracuse used the method of exhaustion to calculate the area under the arc of a parabola with the summation of an infinite series, and gave remarkably accurate approximations of Pi.[18] He also studied the spiral bearing his name and obtained formulas for the volumes of surfaces of revolution.


Woman teaching geometry. Illustration at the beginning of a medieval translation of Euclid's Elements, (c.1310)
In the Middle Ages, mathematics in medieval Islam contributed to the development of geometry, especially algebraic geometry[19][page needed] and geometric algebra.[20] Al-Mahani (b. 853) conceived the idea of reducing geometrical problems such as duplicating the cube to problems in algebra.[21] Thābit ibn Qurra (known as Thebit in Latin) (836–901) dealt with arithmetic operations applied to ratios of geometrical quantities, and contributed to the development of analytic geometry.[22] Omar Khayyám (1048–1131) found geometric solutions to cubic equations.[23] The theorems of Ibn al-Haytham (Alhazen), Omar Khayyam and Nasir al-Din al-Tusi on quadrilaterals, including the Lambert quadrilateral and Saccheri quadrilateral, were early results in hyperbolic geometry, and along with their alternative postulates, such as Playfair's axiom, these works had a considerable influence on the development of non-Euclidean geometry among later European geometers, including Witelo (c.1230–c.1314), Gersonides (1288–1344), Alfonso, John Wallis, and Giovanni Girolamo Saccheri.[24]
In the early 17th century, there were two important developments in geometry. The first was the creation of analytic geometry, or geometry with coordinates and equations, by René Descartes (1596–1650) and Pierre de Fermat (1601–1665). This was a necessary precursor to the development of calculus and a precise quantitative science of physics. The second geometric development of this period was the systematic study of projective geometry by Girard Desargues (1591–1661). Projective geometry is a geometry without measurement or parallel lines, just the study of how points are related to each other.
Two developments in geometry in the 19th century changed the way it had been studied previously. These were the discovery of non-Euclidean geometries by Nikolai Ivanovich Lobachevsky (1792–1856), János Bolyai (1802–1860) and Carl Friedrich Gauss (1777–1855) and of the formulation of symmetry as the central consideration in the Erlangen Programme of Felix Klein (which generalized the Euclidean and non-Euclidean geometries). Two of the master geometers of the time were Bernhard Riemann (1826–1866), working primarily with tools from mathematical analysis, and introducing the Riemann surface, and Henri Poincaré, the founder of algebraic topology and the geometric theory of dynamical systems. As a consequence of these major changes in the conception of geometry, the concept of "space" became something rich and varied, and the natural background for theories as different as complex analysis and classical mechanics.
[edit]Contemporary geometry

[edit]Euclidean geometry


The 421polytope, orthogonally projected into the E8 Lie group Coxeter plane
Euclidean geometry has become closely connected with computational geometry, computer graphics, convex geometry, discrete geometry, and some areas of combinatorics. Momentum was given to further work on Euclidean geometry and the Euclidean groups by crystallography and the work of H. S. M. Coxeter, and can be seen in theories of Coxeter groups and polytopes. Geometric group theory is an expanding area of the theory of more general discrete groups, drawing on geometric models and algebraic techniques.
[edit]Differential geometry
Differential geometry has been of increasing importance to mathematical physics due to Einstein's general relativity postulation that the universe is curved. Contemporary differential geometry is intrinsic, meaning that the spaces it considers are smooth manifolds whose geometric structure is governed by a Riemannian metric, which determines how distances are measured near each point, and not a priori parts of some ambient flat Euclidean space.
[edit]Topology and geometry


A thickening of the trefoil knot
The field of topology, which saw massive development in the 20th century, is in a technical sense a type of transformation geometry, in which transformations are homeomorphisms. This has often been expressed in the form of the dictum 'topology is rubber-sheet geometry'. Contemporary geometric topology and differential topology, and particular subfields such as Morse theory, would be counted by most mathematicians as part of geometry. Algebraic topology and general topology have gone their own ways.
[edit]Algebraic geometry


Quintic Calabi–Yau threefold
The field of algebraic geometry is the modern incarnation of the Cartesian geometry of co-ordinates. From late 1950s through mid-1970s it had undergone major foundational development, largely due to work of Jean-Pierre Serre and Alexander Grothendieck. This led to the introduction of schemes and greater emphasis on topological methods, including various cohomology theories. One of seven Millennium Prize problems, the Hodge conjecture, is a question in algebraic geometry.
The study of low dimensional algebraic varieties, algebraic curves, algebraic surfaces and algebraic varieties of dimension 3 ("algebraic threefolds"), has been far advanced. Gröbner basis theory and real algebraic geometry are among more applied subfields of modern algebraic geometry. Arithmetic geometry is an active field combining algebraic geometry and number theory. Other directions of research involve moduli spaces and complex geometry. Algebro-geometric methods are commonly applied in string and brane theory.
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[edit]References

[edit]Notes
^ Martin J. Turner,Jonathan M. Blackledge,Patrick R. Andrews (1998). "Fractal geometry in digital imaging". Academic Press. p.1. ISBN 0-12-703970-8
^ It is quite common in algebraic geometry to speak about geometry of algebraic varieties over finite fields, possibly singular. From a naïve perspective, these objects are just finite sets of points, but by invoking powerful geometric imagery and using well developed geometric techniques, it is possible to find structure and establish properties that make them somewhat analogous to the ordinary spheres or cones.
^ Kline (1972) "Mathematical thought from ancient to modern times", Oxford University Press, p. 1032. Kant did not reject the logical (analytic a priori) possibility of non-Euclidean geometry, see Jeremy Gray, "Ideas of Space Euclidean, Non-Euclidean, and Relativistic", Oxford, 1989; p. 85. Some have implied that, in light of this, Kant had in fact predicted the development of non-Euclidean geometry, cf. Leonard Nelson, "Philosophy and Axiomatics," Socratic Method and Critical Philosophy, Dover, 1965; p.164.
^ http://www.maths.tcd.ie/pub/HistMath/People/Riemann/Geom/
^ J. Friberg, "Methods and traditions of Babylonian mathematics. Plimpton 322, Pythagorean triples, and the Babylonian triangle parameter equations", Historia Mathematica, 8, 1981, pp. 277—318.
^ Neugebauer, Otto (1969) [1957]. The Exact Sciences in Antiquity (2 ed.). Dover Publications. ISBN 978-0-486-22332-2. Chap. IV "Egyptian Mathematics and Astronomy", pp. 71–96.
^ (Boyer 1991, "Egypt" p. 19)
^ The Journal of Egyptian Archaeology. Vol. 84, 1998 Gnomons at Meroë and Early Trigonometry. pg. 171
^ Neolithic Skywatchers. May 27, 1998 by Andrew L. Slayman Archaeology.org
^ (Boyer 1991, "Ionia and the Pythagoreans" p. 43)
^ Eves, Howard, An Introduction to the History of Mathematics, Saunders, 1990, ISBN 0-03-029558-0.
^ Kurt Von Fritz (1945). "The Discovery of Incommensurability by Hippasus of Metapontum". The Annals of Mathematics.
^ James R. Choike (1980). "The Pentagram and the Discovery of an Irrational Number". The Two-Year College Mathematics Journal.
^ (Boyer 1991, "The Age of Plato and Aristotle" p. 92)
^ (Boyer 1991, "Euclid of Alexandria" p. 119)
^ (Boyer 1991, "Euclid of Alexandria" p. 104)
^ Howard Eves, An Introduction to the History of Mathematics, Saunders, 1990, ISBN 0-03-029558-0 p. 141: "No work, except The Bible, has been more widely used...."
^ O'Connor, J.J. and Robertson, E.F. (February 1996). "A history of calculus". University of St Andrews. Retrieved 2007-08-07.
^ R. Rashed (1994), The development of Arabic mathematics: between arithmetic and algebra, London
^ Boyer (1991). "The Arabic Hegemony". pp. 241–242. "Omar Khayyam (ca. 1050–1123), the "tent-maker," wrote an Algebra that went beyond that of al-Khwarizmi to include equations of third degree. Like his Arab predecessors, Omar Khayyam provided for quadratic equations both arithmetic and geometric solutions; for general cubic equations, he believed (mistakenly, as the 16th century later showed), arithmetic solutions were impossible; hence he gave only geometric solutions. The scheme of using intersecting conics to solve cubics had been used earlier by Menaechmus, Archimedes, and Alhazan, but Omar Khayyam took the praiseworthy step of generalizing the method to cover all third-degree equations (having positive roots). .. For equations of higher degree than three, Omar Khayyam evidently did not envision similar geometric methods, for space does not contain more than three dimensions, ... One of the most fruitful contributions of Arabic eclecticism was the tendency to close the gap between numerical and geometric algebra. The decisive step in this direction came much later with Descartes, but Omar Khayyam was moving in this direction when he wrote, "Whoever thinks algebra is a trick in obtaining unknowns has thought it in vain. No attention should be paid to the fact that algebra and geometry are different in appearance. Algebras are geometric facts which are proved.""
^ O'Connor, John J.; Robertson, Edmund F., "Al-Mahani", MacTutor History of Mathematics archive, University of St Andrews.
^ O'Connor, John J.; Robertson, Edmund F., "Al-Sabi Thabit ibn Qurra al-Harrani", MacTutor History of Mathematics archive, University of St Andrews.
^ O'Connor, John J.; Robertson, Edmund F., "Omar Khayyam", MacTutor History of Mathematics archive, University of St Andrews.
^ Boris A. Rosenfeld and Adolf P. Youschkevitch (1996), "Geometry", in Roshdi Rashed, ed., Encyclopedia of the History of Arabic Science, Vol. 2, p. 447–494 [470], Routledge, London and New York:
"Three scientists, Ibn al-Haytham, Khayyam, and al-Tusi, had made the most considerable contribution to this branch of geometry whose importance came to be completely recognized only in the 19th century. In essence, their propositions concerning the properties of quadrangles which they considered, assuming that some of the angles of these figures were acute of obtuse, embodied the first few theorems of the hyperbolic and the elliptic geometries. Their other proposals showed that various geometric statements were equivalent to the Euclidean postulate V. It is extremely important that these scholars established the mutual connection between this postulate and the sum of the angles of a triangle and a quadrangle. By their works on the theory of parallel lines Arab mathematicians directly influenced the relevant investiagtions of their European counterparts. The first European attempt to prove the postulate on parallel lines – made by Witelo, the Polish scientists of the 13th century, while revising Ibn al-Haytham's Book of Optics (Kitab al-Manazir) – was undoubtedly prompted by Arabic sources. The proofs put forward in the 14th century by the Jewish scholar Levi ben Gerson, who lived in southern France, and by the above-mentioned Alfonso from Spain directly border on Ibn al-Haytham's demonstration. Above, we have demonstrated that Pseudo-Tusi's Exposition of Euclid had stimulated both J. Wallis's and G. Saccheri's studies of the theory of parallel lines."
[edit]Sources

Boyer, C. B. A History of Mathematics, 2nd ed. rev. by Uta C. Merzbach. New York: Wiley, 1989 ISBN 0-471-09763-2 (1991 pbk ed. ISBN 0-471-54397-7).
Nikolai I. Lobachevsky, Pangeometry, Translator and Editor: A. Papadopoulos, Heritage of European Mathematics Series, Vol. 4, European Mathematical Society, 2010.
[edit]Bibliography
Mlodinow, M.; Euclid's window (the story of geometry from parallel lines to hyperspace), UK edn. Allen Lane, 1992.
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Buddhist art
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This article's lead section may not adequately summarize key points of its contents. Please consider expanding the lead to provide an accessible overview of all important aspects of the article. (October 2012)
Buddhist art originated on the Indian subcontinent following the historical life of Siddhartha Gautama, 6th to 5th century BC, and thereafter evolved by contact with other cultures as it spread throughout Asia and the world.
Buddhist art followed believers as the dharma spread, adapted, and evolved in each new host country. It developed to the north through Central Asia and into Eastern Asia to form the Northern branch of Buddhist art, and to the east as far as Southeast Asia to form the Southern branch of Buddhist art. In India, Buddhist art flourished and influenced the development of Hindu art, until Buddhism nearly disappeared in India around the 10th century due in part to the vigorous expansion of Islam alongside Hinduism.
Contents [hide]
1 Pre-iconic phase (5th century - 1st century BCE)
2 Iconic phase (1st century AD – present)
3 Northern Buddhist art
3.1 Afghanistan
3.2 Central Asia
3.3 China
3.3.1 Northern Dynasties
3.3.2 Tang Dynasty
3.3.3 Qing Dynasty
3.3.4 Legacy
3.4 Korea
3.4.1 Three Kingdoms of Korea
3.4.2 Unified Silla
3.4.3 Goryeo Dynasty
3.4.4 Joseon Dynasty
3.5 Japan
3.6 Tibet and Bhutan
3.7 Vietnam
4 Southern Buddhist art
4.1 Sri Lanka
4.2 Myanmar
4.3 Cambodia
4.4 Thailand
4.5 Indonesia
5 Contemporary Buddhist art
6 See also
7 References
8 Bibliography
9 Further reading
10 External links
[edit]Pre-iconic phase (5th century - 1st century BCE)



Footprint of the Buddha. 1st century, Gandhara.
During the 2nd to 1st century BCE, sculptures became more explicit, representing episodes of the Buddha’s life and teachings. These took the form of votive tablets or friezes, usually in relation to the decoration of stupas. Although India had a long sculptural tradition and a mastery of rich iconography, the Buddha was never represented in human form, but only through Buddhist symbolism. This period may have been aniconic.
This reluctance towards anthropomorphic representations of the Buddha, and the sophisticated development of aniconic symbols to avoid it (even in narrative scene where other human figures would appear). This tendency remained as late as the 2nd century CE in the southern parts of India, in the art of the Amaravati School (see: Mara's assault on the Buddha). It has been argued that earlier anthropomorphic representations of the Buddha may have been made of wood and may have perished since then. However, no related archaeological evidence has been found.
The earliest works of Buddhist art in India date back to the 1st century B.C. The Mahabodhi Temple at Bodh Gaya became a model for similar structures in Burma and Indonesia. The frescoes at Sigiriya are said to be even older than the Ajanta Caves paintings.[1]
[edit]Iconic phase (1st century AD – present)

Anthropomorphic representations of the Buddha started to emerge from the 1st century AD in Northern India. The two main centers of creation have been identified as Gandhara in today’s North West Frontier Province, in Pakistan, and the region of Mathura, in central northern India.
The art of Gandhara benefited from centuries of interaction with Greek culture since the conquests of Alexander the Great in 332 BC and the subsequent establishment of the Greco-Bactrian and Indo-Greek Kingdoms, leading to the development of Greco-Buddhist art. Gandharan Buddhist sculpture displays Greek artistic influence, and it has been suggested that the concept of the "man-god" was essentially inspired by Greek mythological culture. Artistically, the Gandharan school of sculpture is said to have contributed wavy hair, drapery covering both shoulders, shoes and sandals, acanthus leaf decorations, etc.
The art of Mathura tends to be based on a strong Indian tradition, exemplified by the anthropomorphic representation of divinities such as the Yaksas, although in a style rather archaic compared to the later representations of the Buddha. The Mathuran school contributed clothes covering the left shoulder of thin muslin, the wheel on the palm, the lotus seat, etc.
Mathura and Gandhara also strongly influenced each other. During their artistic florescence, the two regions were even united politically under the Kushans, both being capitals of the empire. It is still a matter of debate whether the anthropomorphic representations of Buddha was essentially a result of a local evolution of Buddhist art at Mathura, or a consequence of Greek cultural influence in Gandhara through the Greco-Buddhist syncretism.


Representation of the Buddha in the Greco-Buddhist art of Gandhara, 1st century AD.
This iconic art was characterized from the start by a realistic idealism, combining realistic human features, proportions, attitudes and attributes, together with a sense of perfection and serenity reaching to the divine. This expression of the Buddha as both man and God became the iconographic canon for subsequent Buddhist art.
It is interesting to note that the Buddha is an extensively used subject in plastic arts such as sculpture, paintings and literature, but not in music and dance.
Buddhist art continued to develop in India for a few more centuries. The pink sandstone sculptures of Mathura evolved during the Gupta period (4th to 6th century) to reach a very high fineness of execution and delicacy in the modeling. The art of the Gupta school was extremely influential almost everywhere in the rest of Asia. By the 10th century, Buddhist art creation was dying out in India, as Hinduism and Islam ultimately prevailed. At the end of the 12th century A.D. Buddhism in its full glory came to be preserved only in the Himalayan regions in India. These areas, helped by their location, were in greater contact with Tibet and China - for example the art and traditions of Ladakh bear the stamp of Tibetan and Chinese influence.
As Buddhism expanded outside of India from the 1st century AD, its original artistic package blended with other artistic influences, leading to a progressive differentiation among the countries adopting the faith.
A Northern route was established from the 1st century CE through Central Asia, Nepal, Tibet, Bhutan, China, Korea, Japan and Vietnam, in which Mahayana Buddhism prevailed.
A Southern route, where Theravada Buddhism dominated, went through Myanmar, Sri Lanka, Thailand, Cambodia, and Laos.
[edit]Northern Buddhist art



A Chinese wooden Bodhisattva from the Song Dynasty (960-1279 CE)
The Silk Road transmission of Buddhism to Central Asia, China and ultimately Korea and Japan started in the 1st century CE with a semi-legendary account of an embassy sent to the West by the Chinese Emperor Ming (58-75 AD). However, extensive contacts started in the 2nd century CE, probably as a consequence of the expansion of the Kushan Empire into the Chinese territory of the Tarim Basin, with the missionary efforts of a great number of Central Asian Buddhist monks to Chinese lands. The first missionaries and translators of Buddhists scriptures into Chinese, such as Lokaksema, were either Parthian, Kushan, Sogdian or Kuchean.
Central Asian missionary efforts along the Silk Road were accompanied by a flux of artistic influences, visible in the development of Serindian art from the 2nd through the 11th century in the Tarim Basin, modern Xinjiang. Serindian art often derives from the Greco-Buddhist art of the Gandhara district of what is now Pakistan, combining Indian, Greek and Roman influences. Silk Road Greco-Buddhist artistic influences can be found as far as Japan to this day, in architectural motifs, Buddhist imagery, and a select few representations of Japanese gods.
The art of the northern route was also highly influenced by the development of Mahāyāna Buddhism, an inclusive branch of Buddhism characterized by the adoption of new texts, in addition to the traditional āgamas, and a shift in the understanding of Buddhism. Mahāyāna goes beyond the traditional Early Buddhist ideal of the release from suffering (duḥkha) of arhats, and emphasizes the bodhisattva path. The Mahāyāna sutras elevate the Buddha to a transcendent and infinite being, and feature a pantheon of bodhisattvas devoting themselves to the Six Perfections, ultimate knowledge (Prajñāpāramitā), enlightenment, and the liberation of all sentient beings. Northern Buddhist art thus tends to be characterized by a very rich and syncretic Buddhist pantheon, with a multitude of images of the various buddhas, bodhisattvas, and heavenly beings (devas).
[edit]Afghanistan


Statue from a Buddhist monastery, 700 AD, Afghanistan
Buddhist art in Afghanistan (old Bactria) persisted for several centuries until the spread of Islam in the 7th century. It is exemplified by the Buddhas of Bamyan. Other sculptures, in stucco, schist or clay, display very strong blending of Indian post-Gupta mannerism and Classical influence, Hellenistic or possibly even Greco-Roman.
Although Islamic rule was somewhat tolerant of other religions "of the Book", it showed little tolerance for Buddhism, which was perceived as a religion depending on "idolatry". Human figurative art forms also being prohibited under Islam, Buddhist art suffered numerous attacks, which culminated with the systematic destructions by the Taliban regime. The Buddhas of Bamyan, the sculptures of Hadda, and many of the remaining artifacts at the Afghanistan museum have been destroyed.
The multiple conflicts since the 1980s also have led to a systematic pillage of archaeological sites apparently in the hope of reselling in the international market what artifacts could be found.
[edit]Central Asia
Central Asia long played the role of a meeting place between China, India and Persia. During the 2nd century BCE, the expansion of the Former Han to the West led to increased contact with the Hellenistic civilizations of Asia, especially the Greco-Bactrian Kingdom.


Serindian art, 6th-7th century terracotta, Tumshuq (Xinjiang).
Thereafter, the expansion of Buddhism to the North led to the formation of Buddhist communities and even Buddhist kingdoms in the oasis of Central Asia. Some Silk Road cities consisted almost entirely of Buddhist stupas and monasteries, and it seems that one of their main objectives was to welcome and service travelers between East and West.
The eastern part of Central Asia (Chinese Turkestan (Tarim Basin, Xinjiang) in particular has revealed an extremely rich Serindian art (wall paintings and reliefs in numerous caves, portable paintings on canvas, sculpture, ritual objects), displaying multiple influences from Indian and Hellenistic cultures. Works of art reminiscent of the Gandharan style, as well as scriptures in the Gandhari script Kharoshti have been found. These influences were rapidly absorbed however by the vigorous Chinese culture, and a strongly Chinese particularism develops from that point.
See also: Dunhuang, Mogao Caves, Kingdom of Khotan, Silk Road, Silk Road transmission of Buddhism
[edit]China
Buddhism arrived in China around the 1st century AD, and introduced new types of art into China, particularly in the area of statuary. Receiving this distant religion, strong Chinese traits were incorporated into Buddhist art.
[edit]Northern Dynasties


A Chinese Northern Wei Buddha Maitreya, 443 AD.
In the 5th to 6th centuries, the Northern Dynasties, developed rather symbolic and abstract modes of representation, with schematic lines. Their style is also said to be solemn and majestic. The lack of corporeality of this art, and its distance from the original Buddhist objective of expressing the pure ideal of enlightenment in an accessible and realistic manner, progressively led to a change towards more naturalism and realism, leading to the expression of Tang Buddhist art.
Sites preserving Northern Wei Dynasty Buddhist sculpture:
Yungang Grottoes, Shanxi
Longmen Grottoes, Henan
Bingling Temple, Gansu
[edit]Tang Dynasty
Following a transition under the Sui Dynasty, Buddhist sculpture of the Tang evolved towards a markedly lifelike expression. Because of the dynasty's openness to foreign influences, and renewed exchanges with Indian culture due to the numerous travels of Chinese Buddhist monks to India, Tang dynasty Buddhist sculpture assumed a rather classical form, inspired by the Indian art of the Gupta period. During that time, the Tang capital of Chang'an (today's Xi'an) became an important center for Buddhism. From there Buddhism spread to Korea, and Japanese missions to Tang China helped it gain a foothold in Japan.


Tang Bodhisattva.
However, foreign influences came to be negatively perceived in China towards the end of the Tang dynasty. In the year 845, the Tang emperor Wuzong outlawed all "foreign" religions (including Christian Nestorianism, Zoroastrianism and Buddhism) in order to support the indigenous religion, Taoism. He confiscated Buddhist possessions, and forced the faith to go underground, therefore affecting the development of the religion and its arts in China.
Chán Buddhism however, as the origin of Japanese Zen, continued to prosper for some centuries, especially under the Song Dynasty (960-1279), when Chan monasteries were great centers of culture and learning.


Portrait of the Chinese Zen Buddhist Wuzhun Shifan, painted in 1238 AD, Song Dynasty.
Early paintings by Chán monks tended to eschew the meticulous realism of Gongbi painting in favour of vigorous, monochrome paintings, attempting to express the impact of enlightenment through their brushwork.[2]
The rise of Neo-Confucianism under Zhu Xi in the twelfth century resulted in considerable criticism of the monk-painters. Connected as they were with the then-unpopular school of Chan Buddhism, their paintings were discarded and ignored. Some paintings survived after being transported to Japan by visiting Zen monks, but the school of Chan painting gradually diminished.[3]
[edit]Qing Dynasty
Early in the Qing Dynasty, the so-called Four Monk painters (Zhu Da, Shi Tao, Kun Can and Hong Ren) used their paintings to convey their disapproval of the contemporary political climate. Although they used traditional forms, they moved away from the highly technical works popular at the time (exemplified by the Four Wangs) and concentrated on expressive brush stokes and bold colours.[4]
[edit]Legacy
The popularization of Buddhism in China has made the country home to one of the richest collections of Buddhist arts in the world. The Mogao Caves near Dunhuang and the Bingling Temple caves near Yongjing in Gansu province, the Longmen Grottoes near Luoyang in Henan province, the Yungang Grottoes near Datong in Shanxi province, and the Dazu Rock Carvings near Chongqing municipality are among the most important and renowned Buddhist sculptural sites. The Leshan Giant Buddha, carved out of a hillside in the 8th century during the Tang Dynasty and looking down on the confluence of three rivers, is still the largest stone Buddha statue in the world.
[edit]Korea
See also: Buddhism in Korea, Korean Buddhist sculpture, and Korean art
Korean Buddhist art generally reflects an interaction between other Buddhist influences and a strongly original Korean culture. Additionally, the art of the steppes, particularly Siberian and Scythian influences, are evident in early Korean Buddhist art based on the excavation of artifacts and burial goods such as Silla royal crowns, belt buckles, daggers, and comma-shaped gogok.[5][6] The style of this indigenous art was geometric, abstract and richly adorned with a characteristic "barbarian" luxury[clarify]. Although many other influences were strong, Korean Buddhist art, "bespeaks a sobriety, taste for the right tone, a sense of abstraction but also of colours that curiously enough are in line with contemporary taste" (Pierre Cambon, Arts asiatiques- Guimet').
[edit]Three Kingdoms of Korea


Bangasayusang, semi-seated contemplative Maitreya probably from Silla circa early 7th century.
The first of the Three Kingdoms of Korea to officially receive Buddhism was Goguryeo in 372.[7] However, Chinese records and the use of Buddhist motifs in Goguryeo murals indicate the introduction of Buddhism earlier than the official date.[8] The Baekje Kingdom officially recognized Buddhism in 384.[7] The Silla Kingdom, isolated and with no easy sea or land access to China, officially adopted Buddhism in 535 although the foreign religion was known in the kingdom due to the work of Goguryeo monks since the early 5th century.[9] The introduction of Buddhism stimulated the need for artisans to create images for veneration, architects for temples, and the literate for the Buddhist sutras and transformed Korean civilization. Particularly important in the transmission of sophisticated art styles to the Korean kingdoms was the art of the "barbarian" Tuoba, a clan of non-Han Chinese Xianbei people who established the Northern Wei Dynasty in China in 386. The Northern Wei style was particularly influential in the art of the Goguryeo and Baekje. Baekje artisans later transmitted this style along with Southern Dynasty elements and distinct Korean elements to Japan. Korean artisans were highly selective of the styles they incorporated and combined different regional styles together to create a specific Korean Buddhist art style.[10][11]


Seokguram Grotto is a World Heritage Site and dates to the Unified Silla era.
While Goguryeo Buddhist art exhibited vitality and mobility akin with Northern Wei prototypes, the Baekje Kingdom was also in close contact with the Southern Dynasties of China and this close diplomatic contact is exemplified in the gentle and proportional sculpture of the Baekje, epitomized by Baekje sculpture exhibiting the fathomless smile known to art historians as the Baekje smile.[12] The Silla Kingdom also developed a distinctive Buddhist art tradition epitomized by the Bangasayusang, a half-seated contemplative maitreya whose Korean-made twin, the Miroku Bosatsu, was sent to Japan as a proselytizing gift and now resides in the Koryu-ji Temple in Japan.[13] Buddhism in the Three Kingdoms period stimulated massive temple-building projects, such as the Mireuksa Temple in the Baekje Kingdom and the Hwangnyongsa Temple in Silla. Baekje architects were famed for their skill and were instrumental in building the massive nine-story pagoda at Hwangnyongsa and early Buddhist temples in Yamato Japan such as Hōkō-ji (Asuka-dera) and Hōryū-ji.[14] 6th century Korean Buddhist art exhibited the cultural influences of China and India but began to show distinctive indigenous characteristics.[15] These indigenous characteristics can be seen in early Buddhist art in Japan and some early Japanese Buddhist sculpture is now believed to have originated in Korea, particularly from Baekje, or Korean artisans who immigrated to Yamato Japan.[16] Particularly, the semi-seated Maitreya form was adapted into a highly developed Korean style which was transmitted to Japan as evidenced by the Koryu-ji Miroku Bosatsu and the Chugu-ji Siddhartha statues. Although many historians portray Korea as a mere transmitter of Buddhism, the Three Kingdoms, and particularly Baekje, were instrumental as active agents in the introduction and formation of a Buddhist tradition in Japan in 538 or 552.[17]
[edit]Unified Silla


The Goryeo era Gyeongcheonsa Pagoda sits on the first floor of the National Museum of Korea.
During the Unified Silla period, East Asia was particularly stable with China and Korea both enjoying unified governments. Early Unified Silla art combined Silla styles and Baekje styles. Korean Buddhist art was also influenced by new Tang Dynasty styles as evidenced by a new popular Buddhist motif with full-faced Buddha sculptures. Tang China was the cross roads of East, Central, and South Asia and so the Buddhist art of this time period exhibit the so-called international style. State-sponsored Buddhist art flourished during this period, the epitome of which is the Seokguram Grotto.
[edit]Goryeo Dynasty
The fall of the Unified Silla Dynasty and the establishment of the Goryeo Dynasty in 918 indicates a new period of Korean Buddhist art. The Goryeo kings also lavishly sponsored Buddhism and Buddhist art flourished, especially Buddhist paintings and illuminated sutras written in gold and silver ink. [3]. The crowning achievement of this period is the carving of approximately 80,000 woodblocks of the Tripitaka Koreana which was done twice.
[edit]Joseon Dynasty
The Joseon Dynasty actively suppressed Buddhism beginning in 1406 and Buddhist temples and art production subsequently decline in quality in quantity although beginning in 1549, Buddhist art does continue to be produced. [4].
[edit]Japan
See also: Buddhism in Japan, Japanese art, and Buddhist art in Japan


The ASURA in Kōfuku-ji, Nara (734)
Before the introduction of Buddhism, Japan had already been the seat of various cultural (and artistic) influences, from the abstract linear decorative art of the indigenous Neolithic Jōmon from around 10500 BC to 300 BC, to the art during the Yayoi and Kofun periods, with developments such as Haniwa art.
The Japanese discovered Buddhism in the 6th century when missionary monks travelled to the islands together with numerous scriptures and works of art. The Buddhist religion was adopted by the state in the following century. Being geographically at the end of the Silk Road, Japan was able to preserve many aspects of Buddhism at the very time it was disappearing in India, and being suppressed in Central Asia and China.


Scroll calligraphy of Bodhidharma "Zen points directly to the human heart, see into your nature and become Buddha", by Hakuin Ekaku (1686 to 1769)
From 711, numerous temples and monasteries were built in the capital city of Nara, including a five-story pagoda, the Golden Hall of the Horyuji, and the Kōfuku-ji temple. Countless paintings and sculptures were made, often under governmental sponsorship. Indian, Hellenistic, Chinese and Korean artistic influences blended into an original style characterized by realism and gracefulness. The creation of Japanese Buddhist art was especially rich between the 8th and 13th centuries during the periods of Nara, Heian and Kamakura. Japan developed an extremely rich figurative art for the pantheon of Buddhist deities, sometimes combined with Hindu and Shinto influences. This art can be very varied, creative and bold.
From the 12th and 13th, a further development was Zen art, and it faces golden days in Muromachi Period, following the introduction of the faith by Dogen and Eisai upon their return from China. Zen art is mainly characterized by original paintings (such as sumi-e) and poetry (especially haikus), striving to express the true essence of the world through impressionistic and unadorned "non-dualistic" representations. The search for enlightenment "in the moment" also led to the development of other important derivative arts such as the Chanoyu tea ceremony or the Ikebana art of flower arrangement. This evolution went as far as considering almost any human activity as an art with a strong spiritual and aesthetic content, first and foremost in those activities related to combat techniques (martial arts).
Buddhism remains very active in Japan to this day. Still around 80,000 Buddhist temples are preserved. Many of them are in wood and are regularly restored.
[edit]Tibet and Bhutan
See also: Bhutanese art and Tibetan art


Yama 18th century, Tibet
Tantric Buddhism started as a movement in eastern India around the 5th or the 6th century. Many of the practices of Tantric Buddhism are derived from Brahmanism (the usage of mantras, yoga, or the burning of sacrificial offerings). Tantrism became the dominant form of Buddhism in Tibet from the 8th century. Due to its geographical centrality in Asia, Tibetan Buddhist art received influence from Indian, Nepali, Greco-Buddhist and Chinese art.
One of the most characteristic creations of Tibetan Buddhist art are the mandalas, diagrams of a "divine temple" made of a circle enclosing a square, the purpose of which is to help Buddhist devotees focus their attention through meditation and follow the path to the central image of the Buddha. Artistically, Buddhist Gupta art and Hindu art tend to be the two strongest inspirations of Tibetan art.
[edit]Vietnam


The boy Buddha rising up from lotus. Crimson and gilded wood, Trần-Hồ dynasty, Vietnam, 14th-15th century
Chinese influence was predominant in the north of Vietnam (Tonkin) between the 1st and 9th centuries, and Confucianism and Mahayana Buddhism were prevalent. Overall, the art of Vietnam has been strongly influenced by Chinese Buddhist art.
In the south thrived the former kingdom of Champa (before it was later overtaken by the Vietnamese from the north). Champa had a strongly Indianized art, just as neighboring Cambodia. Many of its statues were characterized by rich body adornments. The capital of the kingdom of Champa was annexed by Vietnam in 1471, and it totally collapsed in the 1720s, while Cham people remain an abundant minority across Southeast Asia.
[edit]Southern Buddhist art

The orthodox forms of Buddhism, also known as Southern Buddhism are still practised in Sri Lanka, Myanmar (Burma), Thailand, Laos, and Cambodia. During the 1st century AD, the trade on the overland Silk Road tended to be restricted by the rise of the Parthian empire in the Middle East, an unvanquished enemy of Rome, just as Romans were becoming extremely wealthy and their demand for Asian luxury was rising. This demand revived the sea connections between the Mediterranean Sea and China, with India as the intermediary of choice. From that time, through trade connections, commercial settlements, and even political interventions, India started to strongly influence Southeast Asian countries. Trade routes linked India with southern Burma, central and southern Siam, lower Cambodia and southern Vietnam, and numerous urbanized coastal settlements were established there.


A Cambodian Buddha, 14th century
For more than a thousand years, Indian influence was therefore the major factor that brought a certain level of cultural unity to the various countries of the region. The Pali and Sanskrit languages and the Indian script, together with Mahayana and Theravada Buddhism, Brahmanism and Hinduism, were transmitted from direct contact and through sacred texts and Indian literature such as the Ramayana and the Mahabharata. This expansion provided the artistic context for the development of Buddhist art in these countries, which then developed characteristics of their own.
Between the 1st and 8th centuries, several kingdoms competed for influence in the region (particularly the Cambodian Funan then the Burmese Mon kingdoms) contributing various artistic characteristics, mainly derived from the Indian Gupta style. Combined with a pervading Hindu influence, Buddhist images, votive tablets and Sanskrit inscriptions are found throughout the area.
From the 9th to the 13th centuries, Southeast Asia had very powerful empires and became extremely active in Buddhist architectural and artistic creation. The Sri Vijaya Empire to the south and the Khmer Empire to the north competed for influence, but both were adherents of Mahayana Buddhism, and their art expressed the rich Mahayana pantheon of the Bodhisattvas. The Theravada Buddhism of the Pali canon was introduced to the region around the 13th century from Sri Lanka, and was adopted by the newly founded ethnic Thai kingdom of Sukhothai. Since in Theravada Buddhism of the period, Monasteries typically were the central places for the laity of the towns to receive instruction and have disputes arbitrated by the monks, the construction of temple complexes plays a particularly important role in the artistic expression of Southeast Asia from that time.
From the 14th century, the main factor was the spread of Islam to the maritime areas of Southeast Asia, overrunning Malaysia, Indonesia, and most of the islands as far as the Philippines. In the continental areas, Theravada Buddhism continued to expand into Burma, Laos and Cambodia.
[edit]Sri Lanka
According to tradition, Buddhism was introduced in Sri Lanka in the 3rd century BC by Indian missionaries under the guidance of Thera Mahinda, the son of the Mauryan Emperor Asoka. Prior to the expansion of Buddhism, the indigenous population of Sri Lanka lived in an animistic world full of superstition. The assimilation and conversion of the various pre-Buddhist beliefs was a slow process. In order to gain a foothold among the rural population, Buddhism needed to assimilate the various categories of spirits and other supernatural beliefs. The earliest monastic complex was the Mahāvihāra at Anurādhapura founded by Devānampiyatissa and presented to Mahinda Thera. The Mahāvihāra became the centre of the orthodox Theravāda doctrine and its supreme position remained unchallenged until the foundation of the Abhayagiri Vihāra around BC 89 by Vaţţagāmaņĩ. The Abhayagiri Vihāra became the seat of the reformed Mahāyāna doctrines. The rivalry between the monks of the Mahāvihāra and the Abhayagiri led to a further split and the foundation of the Jetavanarama near the Mahāvihāra. The main feature of Sinhala Buddhism was its division into three major groups, or nikāyas, named after the three main monastic complexes at Anurādhapura; the Mahāvihāra, the Abhayagiri, and the Jetavanārāma. This was the result in the deviations in the disciplinary rules (vinaya) and doctrinal disputes. All the other monasteries of Sri Lanka owed ecclesiastical allegiance to one of the three. Sri Lanka is famous for its creations of Buddhist sculptures made of stone and cast in bronze alloy.[18]
[edit]Myanmar


A Mandalay-style statue of Buddha
A neighbor of India, Myanmar (Burma) was naturally strongly influenced by the eastern part of Indian territory. The Mon of southern Burma are said to have been converted to Buddhism around 200 BC under the proselytizing of the Indian king Ashoka, before the schism between Mahayana and Hinayana Buddhism.
Early Buddhist temples are found, such as Beikthano in central Myanmar, with dates between the 1st and the 5th centuries. The Buddhist art of the Mons was especially influenced by the Indian art of the Gupta and post-Gupta periods, and their mannerist style spread widely in Southeast Asia following the expansion of the Mon Empire between the 5th and 8th centuries.
Later, thousands of Buddhist temples were built at Bagan, the capital, between the 11th and 13th centuries, and around 2,000 of them are still standing. Beautiful jeweled statues of the Buddha are remaining from that period. Creation managed to continue despite the seizure of the city by the Mongols in 1287.
During the Ava period, from the 14th to 16th centuries, the Ava (Innwa) style of the Buddha image was popular. In this style, the Buddha has large protruding ears, exaggerated eyebrows that curve upward, half-closed eyes, thin lips and a hair bun that is pointed at the top, usually depicted in the bhumisparsa mudra.[19]
During the Konbaung dynasty, at the end of the 18th century, the Mandalay style of the Buddha image emerged, a style that remains popular to this day.[20] There was a marked departure from the Innwa style, and the Buddha's face is much more natural, fleshy, with naturally-slanted eyebrows, slightly slanted eyes, thicker lips, and a round hair bun at the top. Buddha images in this style can be found reclining, standing or sitting.[21] Mandalay-style Buddhas wear flowing, draped robes.
Another common style of Buddha images is the Shan style, from the Shan people, who inhabit the highlands of Myanmar. In this style, the Buddha is depicted with angular features, a large and prominently pointed nose, a hair bun tied similar to Thai styles, and a small, thin mouth.[22]
[edit]Cambodia


Bodhisattva Lokesvara, Cambodia 12th century.
Cambodia was the center of the Funan kingdom, which expanded into Burma and as far south as Malaysia between the 3rd and 6th centuries. Its influence seems to have been essentially political, most of the cultural influence coming directly from India.
Later, from the 9th to 13th centuries, the Mahayana Buddhist and Hindu Khmer Empire dominated vast parts of the Southeast Asian peninsula, and its influence was foremost in the development of Buddhist art in the region. Under the Khmer, more than 900 temples were built in Cambodia and in neighboring Thailand.
Angkor was at the center of this development, with a Buddhist temple complex and urban organization able to support around 1 million urban dwellers. A great deal of Cambodian Buddhist sculpture is preserved at Angkor; however, organized looting has had a heavy impact on many sites around the country.
Often, Khmer art manages to express intense spirituality through divinely beaming expressions, in spite of spare features and slender lines.
See also: Khmer art, Khmer sculpture
[edit]Thailand


Phra Atchana Wat Si Chum, Sukhothai Province, Thailand
From the 1st to the 7th centuries, Buddhist art in Thailand was first influenced by direct contact with Indian traders and the expansion of the Mon kingdom, leading to the creation of Hindu and Buddhist art inspired from the Gupta tradition, with numerous monumental statues of great virtuosity.
From the 9th century, the various schools of Thai art then became strongly influenced by Cambodian Khmer art in the north and Sri Vijaya art in the south, both of Mahayana faith. Up to the end of that period, Buddhist art is characterized by a clear fluidness in the expression, and the subject matter is characteristic of the Mahayana pantheon with multiple creations of Bodhisattvas.
From the 13th century, Theravada Buddhism was introduced from Sri Lanka around the same time as the ethnic Thai kingdom of Sukhothai was established. The new faith inspired highly stylized images in Thai Buddhism, with sometimes very geometrical and almost abstract figures.
During the Ayutthaya period (14th-18th centuries), the Buddha came to be represented in a more stylistic manner with sumptuous garments and jeweled ornamentations. Many Thai sculptures or temples tended to be gilded, and on occasion enriched with inlays.
See also: Thai art, Buddha images in Thailand
[edit]Indonesia


A Buddha in Borobudur.
Like the rest of Southeast Asia, Indonesia seems to have been most strongly influenced by India from the 1st century AD. The islands of Sumatra and Java in western Indonesia were the seat of the empire of Sri Vijaya (8th-13th century), which came to dominate most of the area around the Southeast Asian peninsula through maritime power. The Sri Vijayan Empire had adopted Mahayana and Vajrayana Buddhism, under a line of rulers named the Sailendra. Sri Vijaya spread Mahayana Buddhist art during its expansion into the Southeast Asian peninsula. Numerous statues of Mahayana Bodhisattvas from this period are characterized by a very strong refinement and technical sophistication, and are found throughout the region.


A detailed carved relief stone from Borobudur.


The statue of Prajñāpāramitā from Singhasari, East Java.
Extremely rich and refined architectural remains are found in Java and Sumatra. The most magnificent is the temple of Borobudur (the largest Buddhist structure in the world, built around 780-850 AD). This temple is modelled after the Buddhist concept of universe, the Mandala which counts 505 images of the seated Buddha and unique bell-shaped stupa that contains the statue of Buddha. Borobudur is adorned with long series of bas-reliefs narrated the holy Buddhist scriptures. The oldest Buddhist structure in Indonesia probably is the Batujaya stupas at Karawang, West Java, dated from around 4th century. This temple is some plastered brick stupas. However, Buddhist art in Indonesia reach the golden era during the Sailendra dynasty rule in Java. The bas-reliefs and statues of Boddhisatva, Tara, and Kinnara found in Kalasan, Sewu, Sari, and Plaosan temple is very graceful with serene expression, While Mendut temple near Borobudur, houses the giant statue of Vairocana, Avalokitesvara, and Vajrapani.
In Sumatra Sri Vijaya probably built the temple of Muara Takus, and Muaro Jambi. The most beautiful example of classical Javanese Buddhist art is the serene and delicate statue of Prajnaparamita (the collection of National Museum Jakarta) the goddess of transcendental wisdom from Singhasari kingdom. The Indonesian Buddhist Empire of Sri Vijaya declined due to conflicts with the Chola rulers of India, then followed by Majapahit empire, before being destabilized by the Islamic expansion from the 13th century.
[edit]Contemporary Buddhist art

Many contemporary artists have made use of Buddhist themes. Notable examples are Bill Viola, in his video installations,[23] John Connell, in sculpture.,[24] and Allan Graham in his multi-media “Time is Memory”.[25]
In the UK The Network of Buddhist Organisations has interested itself in identifying Buddhist practitioners across all the arts. In 2005 it co-ordinated the UK-wide Buddhist arts festival, "A Lotus in Flower";[26] in 2009 it helped organise the two-day arts conference, "Buddha Mind, Creative Mind".[27] As a result of the latter an association of Buddhist artists was formed.[28]
[edit]See also

Buddhism
Buddhist architecture
Buddhist music
Buddhist symbolism
Buddharupa
Daibutsu
Borobudur
Eastern art history
[edit]References

James Huntley Grayson (2002). Korea: A Religious History. UK: Routledge. ISBN 0-7007-1605-X.
Gibson, Agnes C. (Tr. from the 'Handbook' of Prof. Albert Grunwedel); Revised and Enlarged by Jas.Burgess (1901). Buddhist Art in India. Bernard Quaritch, London.
^ Buddhist Art Frontline Magazine May 13–26, 1989
^ Cotterell, A; The imperial capitals of China: an inside view of the celestial empire, Random House 2008, ISBN 978-1-84595-010-1 p179
^ Ortiz, Valérie Malenfer; Dreaming the southern song landscape: the power of illusion in Chinese painting, BRILL 1999, ISBN 978-90-04-11011-3 p161-2
^ Confucius Institute Online: Four Wangs and Four Monks
^ "Crown". Arts of Korea. The Metropolitan Museum of Art. Retrieved 2007-01-09.
^ Grayson (2002), p. 21.
^ a b Grayson (2002), p. 25.
^ Grayson (2002), p. 24.
^ Peter N. Stearns and William Leonard Langer (2001). "The Encyclopedia of World History: ancient, medieval, and modern, chronologically arranged". The Encyclopedia of World History: ancient, medieval, and modern, chronologically arranged. Houghton Mifflin Books. ISBN 0-395-65237-5.; "Korea, 500–1000 A.D.". Timeline of Arts History. The Metropolitan Museum of Art. Retrieved 2007-01-09.
^ Grayson (2002), pp. 27 & 33.
^ "Korean Buddhist Sculpture, 5th–9th Century". Timeline of Arts History. The Metropolitan Museum of Art. Retrieved 2007-01-09.
^ metmuseum.org
^ kenyon.edu
^ [1]; orientalarchitecture.com; indiana.edu
^ metmuseum.org
^ buddhapia.com
^ [2]
^ von Schroeder, Ulrich. 1990. Buddhist Sculptures of Sri Lanka. First comprehensive monograph on the stylistic and iconographic development of the Buddhist sculptures of Sri Lanka. 752 pages with 1620 illustrations (20 colour and 1445 half-tone illustrations; 144 drawings and 5 maps. (Hong Kong: Visual Dharma Publications, Ltd.). von Schroeder, Ulrich. 1992. The Golden Age of Sculpture in Sri Lanka - Masterpieces of Buddhist and Hindu Bronzes from Museums in Sri Lanka, [catalogue of the exhibition held at the Arthur M. Sackler Gallery, Washington, D. C., 1st November 1992 – 26th September 1993]. (Hong Kong: Visual Dharma Publications, Ltd.).
^ http://www.seasite.niu.edu/burmese/cooler/Chapter_4/Part1/post_pagan_period__part_1.htm
^ http://www.seasite.niu.edu/burmese/cooler/Chapter_4/Part3/post_pagan_period__part_3.htm
^ http://www.buddhaartgallery.com/mandalay_buddha_statues.html
^ http://www.buddhaartgallery.com/shan_buddha_statues.html
^ Buddha Mind in Contemporary Art, University of California Press, 2004
^ ARTlines, April 1983
^ The Brooklyn Rail, December 2007
^ a poster advertising one of the events is archived here - http://www.nbo.org.uk/whats%20on/poster.pdf
^ http://fwbo-news.blogspot.com/2009/07/report-from-buddha-mind-creative-mind.html
^ http://dharmaarts.ning.com/
[edit]Bibliography

von Schroeder, Ulrich. (1990). Buddhist Sculptures of Sri Lanka. (752 p.; 1620 illustrations). Hong Kong: Visual Dharma Publications, Ltd. ISBN 962-7049-05-0
von Schroeder, Ulrich. (1992). The Golden Age of Sculpture in Sri Lanka - Masterpieces of Buddhist and Hindu Bronzes from Museums in Sri Lanka, [catalogue of the exhibition held at the Arthur M. Sackler Gallery, Washington, D. C., 1 November 1992 – 26 September 1993]. Hong Kong: Visual Dharma Publications, Ltd. ISBN 962-7049-06-9
[edit]Further reading

Foltz, Richard C. (2010). Religions of the Silk Road: Premodern Patterns of Globalization. New York, New York, USA: Palgrave Macmillan. ISBN [[Special:BookSources/978230621251|978230621251]].
Jarrige, Jean-François (2001). Arts asiatiques- Guimet (Éditions de la Réunion des Musées Nationaux ed.). Paris. ISBN 2-7118-3897-8.
Lee, Sherman (2003). A History of Far Eastern Art (5th Edition). New York: Prentice Hall. ISBN 0-13-183366-9.
Scarre, Dr. Chris (editor) (1991). Past Worlds. The Times Atlas of Archeology. London: Times Books Limited. ISBN 0-7230-0306-8.
Susan L. Huntington: "Early Buddhist art and the theory of aniconism", Art Journal, Winter 1990.
D. G. Godse's writings in Marathi.
von Schroeder, Ulrich. 1981. Indo-Tibetan Bronzes. (Hong Kong: Visual Dharma Publications, Ltd.).
von Schroeder, Ulrich. 2001. Buddhist Sculptures in Tibet. Vol. One: India & Nepal; Vol. Two: Tibet & China. (Hong Kong: Visual Dharma Publications, Ltd.).
[edit]External links

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Buddhist architecture
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Buddhist religious architecture developed in South Asia in the 3rd century BCE.
Three types of structures are associated with the religious architecture of early Buddhism: monasteries (viharas), stupas, and temples (Chaitya grihas).
Viharas initially were only temporary shelters used by wandering monks during the rainy season, but later were developed to accommodate the growing and increasingly formalised Buddhist monasticism. An existing example is at Nalanda (Bihar). A distinctive type of fortress architecture found in the former and present Buddhist kingdoms of the Himalayas are dzongs.
The initial function of a stupa was the veneration and safe-guarding of the relics of the Buddha. The earliest surviving example of a stupa is in Sanchi (Madhya Pradesh).
In accordance with changes in religious practice, stupas were gradually incorporated into chaitya-grihas (temple halls). These reached their high point in the 1st century BC, exemplified by the cave complexes of Ajanta and Ellora (Maharashtra). The Mahabodhi Temple at Bodh Gaya in Bihar is another well known example.
The Pagoda is an evolution of the Indian stupa.
Contents [hide]
1 Early development
2 Examples
3 See also
4 References
5 External links
[edit]Early development

Buddhist architecture emerged slowly in the period following the Buddha’s life, building on Brahmanist Vedic models, but incorporating specifically Buddhist symbols.
Brahmanist temples at this time followed a simple plan – a square inner space, the sacrificial arena, often with a surrounding ambulatory route separated by lines of columns, with a conical or rectangular sloping roof, behind a porch or entrance area, generally framed by freestanding columns or a colonnade. The external profile represents Mount Meru, the abode of the gods and centre of the universe. The dimensions and proportions were dictated by sacred mathematical formulae. This simple plan was adopted by early Buddhists, sometimes adapted with additional cells for monks at the periphery (especially in the early cave temples such as at Ajanta, India).[1]
In essence the basic plan survives to this day in Buddhist temples throughout the world. The profile became elaborated and the characteristic mountain shape seen today in many Hindu temples was used in early Buddhist sites and continued in similar fashion in some cultures (such as the Khmer). In others, such as Japan and Thailand, local influences and differing religious practices led to different architecture.
Early temples were often timber, and little trace remains, although stone was increasingly used. Cave temples such as those at Ajanta have survived better and preserve the plan form, porch and interior arrangements from this early period. As the functions of the monastery-temple expanded, the plan form started to diverge from the Brahmanist tradition and became more elaborate, providing sleeping, eating and study accommodation.
A characteristic new development at religious sites was the stupa. Stupas were originally more sculpture than building, essentially markers of some holy site or commemorating a holy man who lived there. Later forms are more elaborate and also in many cases refer back to the Mount Meru model.
One of the earliest Buddhist sites still in existence is at Sanchi, India, and this is centred on a stupa said to have been built by King Ashoka (273-236 BCE). The original simple structure is encased in a later, more decorative one, and over two centuries the whole site was elaborated upon. The four cardinal points are marked by elaborate stone gateways.[2]
As with Buddhist art, architecture followed the spread of Buddhism throughout south and east Asia and it was the early Indian models that served as a first reference point, even though Buddhism virtually disappeared from India itself in the 10th century.
Decoration of Buddhist sites became steadily more elaborate through the last two centuries BCE, with the introduction of tablets and friezes, including human figures, particularly on stupas. However, the Buddha was not represented in human form until the 1st century CE. Instead, aniconic symbols were used. This is treated in more detail in Buddhist art, Aniconic phase. It influenced the development of temples, which eventually became a backdrop for Buddha images in most cases.
As Buddhism spread, Buddhist architecture diverged in style, reflecting the similar trends in Buddhist art. Building form was also influenced to some extent by the different forms of Buddhism in the northern countries, practicing Mahayana Buddhism in the main and in the south where Theravada Buddhism prevailed.
[edit]Examples


Jetavanaramaya stupa is an example of brick-clad Buddhist architecture in Sri Lanka



The Rinpung Dzong follows a distinctive type of fortress architecture found in the former and present Buddhist kingdoms of the Himalayas, most notably Bhutan



The Great Stupa in Sanchi, India is considered a cornerstone of Buddhist architecture



Vatadage Temple, in Polonnaruwa, is a uniquely Sri Lankan circular shrine enclosing a small dagoba. The vatadage has a three-tiered conical roof, spanning a height of 40–50 feet, without a center post, and supported by pillars of diminishing height ]

[edit]See also

Thai temple art and architecture
[edit]References

^ The World of Buddhism, Thames and Hudson, quoted at the Buddhamind website: http://www.buddhamind.info/leftside/arty/architect.htm">http://www.buddhamind.info/leftside/arty/architect.htm
^ Some photographs and further description at http://www.buddhamind.info/leftside/arty/architect.htm">http://www.buddhamind.info/leftside/arty/architect.htm
Wikimedia Commons has media related to: Buddhist architecture
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Peabody Essex Museum—Phillips Library: The Herbert Offen Research Collection — books and items on Buddhist architecture.

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Buddhist symbolism
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This article does not cite any references or sources. Please help improve this article by adding citations to reliable sources. Unsourced material may be challenged and removed. (July 2008)


The eight-spoked Dharmacakra. The eight spokes represent the Noble Eightfold Path of Buddhism.
Buddhist symbolism is the use of Buddhist art to see others represent certain aspects of dhamma, which began in the 4th century BCE. Anthropomorphic symbolism appeared from around the 1st century CE with the arts of Mathura and the Greco-Buddhist art of Gandhara, and were combined with the previous symbols. Various symbolic innovations were later introduced, especially through Tibetan Buddhism.
Contents [hide]
1 Early symbols
2 Mahayana symbolism
3 Theravada symbolism
4 Modern Pan-Buddhist symbolism
5 See also
6 External links
[edit]Early symbols



Footprint of the Buddha. 1st century, Gandhara, with depictions of the triratna and the Dharmacakra.
It is not known what the role of the image was in Early Buddhism, although many surviving images can be found, because their symbolic or representative nature was not clearly explained in early texts. Among the earliest and most common symbols of Buddhism are the stupa, Dharma wheel, and the lotus flower. The dharma wheel, traditionally represented with eight spokes, can have a variety of meanings. It initially only meant royalty (concept of the "Monarch of the Wheel, or Chakravatin), but started to be used in a Buddhist context on the Pillars of Ashoka during the 3rd century BC. The Dharma wheel is generally seen as referring to the historical process of teaching the buddhadharma; the eight spokes refer to the Noble Eightfold Path. The lotus, as well, can have several meanings, often referring to the inherently pure potential of the mind.
Other early symbols include the trisula, a symbol use since around the 2nd century BC that combine the lotus, the vajra diamond rod and a symbolization of the three jewels (The Buddha, the dharma, the sangha). The swastika was traditionally used in India by Buddhists and Hindus as a good luck sign. In East Asia, the swastika is often used as a general symbol of Buddhism. Swastikas used in this context can either be left or right-facing.
Early Buddhism did not portray the Buddha himself and may have been aniconic. The first hint of a human representation in Buddhist symbolism appear with the Buddha footprint......
[edit]Mahayana symbolism

Main article: Ashtamangala


Mahayana Symbolism : Ashtamangala
In Mahayana, Buddhist figures and sacred objects leaned towards esoteric and symbolic meaning. The Mudras are a series of symbolic hand gestures describing the actions of the characters represented in only the most interesting Buddhist art. Many images also function as mandalas.
Mahayana and Vajrayana Buddhist art frequently makes use of a particular set of eight auspicious symbols, ashtamangala, in household and public art. These symbols have spread with Buddhism to many cultures' arts, including Indian, Tibetan, Nepalese, and Chinese art.
These symbols are:
Lotus flower. Representing purity and enlightenment.
Endless knot, or, the Mandala. Representing eternal harmony.
Golden Fish pair. Representing conjugal happiness and freedom.
Victory Banner. Representing a victorious battle.
Wheel of Dharma or Chamaru in Nepali Buddhism. Representing knowledge.
Treasure Vase. Representing inexhaustible treasure and wealth.
Parasol. Representing the crown, and protection from the elements.
Conch shell. Representing the thoughts of the Buddha.
[edit]Theravada symbolism

Main article: Cetiya
In Theravada, Buddhist art stayed strictly in the realm of representational and historic meaning. Reminders of the Buddha, cetiya, were divided up into relic, spatial, and representational memorials.
Although the Buddha was not represented in human form until around the 1st century AD (see Buddhist art), the Physical characteristics of the Buddha are described in one of the central texts of the traditional Pali canon, the Digha Nikaya, in the discourse titled "Sutra of the Marks" (Pali: Lakkhana Sutta) (D.iii.142ff.).
These characteristics comprise 32 signs, "The 32 signs of a Great Man" (Pali: Lakkhana Mahapurisa 32), and were supplemented by another 80 Secondary Characteristics (Pali:Anubyanjana). These traits are said to have defined the appearance of the historical Buddha, Siddhartha Gautama and have been used symbolically in many of his representations.
[edit]Modern Pan-Buddhist symbolism



The international Buddhist flag.
At its founding in 1952, the World Fellowship of Buddhists adopted two symbols [1]. These were a traditional eight-spoked Dharma wheel and the five-colored flag which had been designed in Sri Lanka in the 1880s with the assistance of Henry Steel Olcott [2].
[edit]See also

Buddhist art
Chinese art
Indian art
Japanese art
Korean art
Religious symbolism
Tibetan art
[edit]External links

web site showing iconic representations of the 8 auspicious symbols along with explanations
the eight auspicious symbols of Buddhism — a study in spiritual evolution
General Buddhist Symbols
Tibetan Buddhist Symbols
Buddhist Tantric Symbols
web site showing hand carvings on metal crafts of the 8 auspicious symbols along with explanations
Buddhist Symbols: the Eight Auspicious Signs
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Religious symbolism
From Wikipedia, the free encyclopedia


Symbols from twelve world religious movements.
Row 1: Bahá'í Faith, Buddhism
Row 2: Christianity, Chinese folk religion, Hinduism, Islam
Row 3: Jainism, Judaism, Paganism, Shinto
Row 4: Sikhism, Taoism
Religious symbolism is the use of symbols, including archetypes, acts, artwork, events, or natural phenomena, by a religion. Religions view religious texts, rituals, and works of art as symbols of compelling ideas or ideals. Symbols help create a resonant mythos expressing the moral values of the society or the teachings of the religion, foster solidarity among adherents, and bring adherents closer to their object of worship.
The study of religious symbolism is either universalist, as a component of comparative religion and mythology, or in localized scope, within the confines of a religion's limits and boundaries.
Religious symbolism is effective when it appeals to both the intellect and the emotions. The choice of suitable acts and objects for symbolism is narrow enough that it would not be easy to avoid the appearance of an imitation of other traditions, even if there had been a deliberate attempt to invent an entirely new ritual.[1]
Contents [hide]
1 Religious symbols
2 See also
3 References
4 External links
[edit]Religious symbols

Religion or philosophy Name Symbol
Ayyavazhi
Main article: Ayyavazhi symbolism
Lotus Carrying Namam

Bahá'í Faith
Main article: Bahá'í symbols
Nine Pointed Star

Ringstone

Buddhism
Main article: Buddhist symbolism
Wheel of Dharma

Lotus Flower

Christianity
Main article: Christian symbolism
Christian cross

Ichthys (fish)

Alpha and Omega

Chi Rho

Church of Jesus Christ of Latter-day Saints
Main article: Mormon Symbolism
Angel Moroni (a popular symbol for Mormonism)

Cao Dai Eye of Providence

Contemporary Paganism Ankh (Ancient Egyptian religion and Kemetism)

Arevakhach (Armenian, Hetanism)

Hands of God (Rodnovery a.k.a. Slavic Neopaganism)

Mjolnir (Heathenism aka Germanic Neopaganism)

Kappu ("palm of hand"), a symbol of Natib Qadish.

Pentagram

Pentacle (Wicca)

Triskelion (Celtic Neopaganism)

Triple Goddess (Neopaganism)

Discordianism The Sacred Chao

Eckankar Eckankar symbol

Gnosticism Sun cross

Ouroboros (also a symbol of Alchemy and Hermeticism)

Humanism Happy Human

Hinduism
Main article: Hindu iconography
Pranava

Lotus Flower

Swastika

Islam
Main article: Symbols of Islam
Kalima/Shahadah

Name of Allah

Crescent ("alem", symbol used on top of a minaret)

Rub el Hizb

Jainism Jain symbol

Ahimsa Hand

Judaism
Main article: Jewish symbolism
Star of David

Menorah

The Raëlian movement Star of David intertwined with a Swastika or Swirling Galaxy Star

Rastafari movement Lion of Judah often shown upon the Imperial Ethiopian flag

Ravidassia Harr Nishan

LaVeyan Satanism Sigil of Baphomet

Theistic Satanism Satanic Cross

Sigil of Lucifer (Satan's Seal)

Serer religion Yoonir

Ndut

Sikhism Khanda

Ik Onkar

Shinto Torii

Southeastern Ceremonial Complex Solar cross

Taoism (Daoism) Yin and yang (Taiji)

Thelema Unicursal Hexagram

Universal Sufism Tughra Inayati

Unitarian Universalism Flaming chalice

Zoroastrianism Faravahar

A number of these are represented in Unicode.
[edit]See also

Religion in national symbols
French law on secularity and conspicuous religious symbols in schools
Religious symbolism of unity of opposites
United States military chaplain symbols
[edit]References

United States Veteran's Administration approved religious symbols for graves
^ Baer, Hans A. (1998). William H. Swatos, Jr. ed. Encyclopedia of Religion and Society - Entry for Symbols. Walnut Creek, CA, USA: AltaMira Press. p. 504. ISBN 0761989560. Retrieved 31 October 2008.
[edit]External links

Media related to Religious symbol at Wikimedia Commons
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Tibetan art
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15/16th century carved manuscript cover. An excellent example of the Tibetan carvers art with simple designs containing sacred elements. Sculpted and lacquered, this manuscript cover has stupas and canopies within geometric designs. Primary to this woodcarvings rich appointment of iconography, are the auspicious symbols (ashtamangala) including: the "Precious Umbrella" that symbolizes the wholesome activity of preserving beings from harmful forces; the "Victory Banner" that celebrates the activities of one's own and others' body and mind over obstacles, as well as the "Vase of Treasure" holding and endless reign of wealth and prosperity.
Tibetan art refers to the art of Tibet. For more than a thousand years, Tibetan artists have played a key role in the cultural life of Tibet. From designs for painted furniture to elaborate murals in religious buildings, their efforts have permeated virtually every facet of life on the Tibetan plateau. The vast majority of surviving artworks created before the mid-20th century are dedicated to the depiction of religious subjects, for the most part being distemper on cloth or murals. They were commissioned by religious establishments or by pious individuals for use within the practice of Tibetan Buddhism and were manufactured in large workshops by uncredited artists.
The art of Tibet may be studied in terms of influences which have contributed to it over the centuries, from other Chinese, Nepalese, Indian, and sacred styles.
Contents [hide]
1 Mahayana Buddhist influence
2 Tantric influence
3 Bön influence
4 See also
5 References
6 Contemporary Tibetan art
7 External links
[edit]Mahayana Buddhist influence

As Mahayana Buddhism emerged as a separate school in the 4th century BC it emphasized the role of bodhisattvas, compassionate beings who forego their personal escape to Nirvana in order to assist others. From an early time various bodhisattvas were also subjects of statuary art. Tibetan Buddhism, as an offspring of Mahayana Buddhism, inherited this tradition. But the additional dominating presence of the Vajrayana (or Buddhist tantra) may have had an overriding importance in the artistic culture. A common bodhisattva depicted in Tibetan art is the deity Chenrezig (Avalokitesvara), often portrayed as a thousand-armed saint with an eye in the middle of each hand, representing the all-seeing compassionate one who hears our requests. This deity can also be understood as a Yidam, or 'meditation Buddha' for Vajrayana practice
[edit]Tantric influence



Dharmapala, Field Museum, Chicago.
More specifically, Tibetan Buddhism contains Tantric Buddhism, also known as Vajrayana Buddhism for its common symbolism of the vajra, the diamond thunderbolt (known in Tibetan as the dorje). Most of the typical Tibetan Buddhist art can be seen as part of the practice of tantra. Vajrayana techniques incorporate many visualizations/imaginations during meditation, and most of the elaborate tantric art can be seen as aids to these visualizations; from representations of meditational deities (yidams) to mandalas and all kinds of ritual implements.
A surprising aspect of Tantric Buddhism is the common representation of wrathful deities, often depicted with angry faces, circles of flame, or with the skulls of the dead. These images represent the Protectors (Skt. dharmapala) and their fearsome bearing belies their true compassionate nature. Actually their wrath represents their dedication to the protection of the dharma teaching as well as to the protection of the specific tantric practices to prevent corruption or disruption of the practice. They are most importantly used as wrathful psychological aspects that can be used to conquer the negative attitudes of the practitioner.
[edit]Bön influence

The indigenous shamanistic religion of the Himalayas is known as Bön. Bon contributes a pantheon of local tutelary deities to Tibetan art. In Tibetan temples (known as lhakhang), statues of the Buddha or Padmasambhava are often paired with statues of the tutelary deity of the district who often appears angry or dark. These gods once inflicted harm and sickness on the local citizens but after the arrival of Padmasambhava these negative forces have been subdued and now must serve Buddha.
[edit]See also

Chorten
Dzong architecture
Eight auspicious symbols
Rubin Museum of Art
Sandpainting
Tibetan rugs
Tibetan tsakli
Tibetan Buddhist Wall Paintings
Iron Man, Tibetan Buddhist statue carved from a meteorite
[edit]References

von Schroeder, Ulrich. 1981. Indo-Tibetan Bronzes. (608 pages, 1244 illustrations). Hong Kong: Visual Dharma Publications Ltd. ISBN 962-7049-01-8
von Schroeder, Ulrich. 2001. Buddhist Sculptures in Tibet. Vol. One: India & Nepal; Vol. Two: Tibet & China. (Volume One: 655 pages with 766 illustrations; Volume Two: 675 pages with 987 illustrations). Hong Kong: Visual Dharma Publications, Ltd.). ISBN 962-7049-07-7
von Schroeder, Ulrich. 2006. Empowered Masters: Tibetan Wall Paintings of Mahasiddhas at Gyantse. (p. 224 pages with 91 colour illustrations). Chicago: Serindia Publications. ISBN 1-932476-24-5
von Schroeder, Ulrich. 2008. 108 Buddhist Statues in Tibet. (212 p., 112 colour illustrations) (DVD with 527 digital photographs). Chicago: Serindia Publications. ISBN 962-7049-08-5

[edit]Contemporary Tibetan art

Contemporary Tibetan art refers to the art of modern Tibet, or Tibet after 1950. It can also refer to art by the Tibetan diaspora, which is explicitly political and religious in nature. Contemporary Tibetan art includes modern thangka (religious scroll paintings) that resemble ancient thangka, as well as radical, avant-garde, works.
[edit]External links

Museum exhibit of Tibetan art.
Mechak Center for Contemporary Tibetan Art
Peak Art Gallery, A Contemporary Tibetan Art Gallery
Art of Tibet An online portal about eastern culture, art, the Dharma, and more.
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Dzong architecture
From Wikipedia, the free encyclopedia


Rinpung Dzong at Paro, with watchtower seen above.
This article contains Indic text. Without proper rendering support, you may see question marks or boxes, misplaced vowels or missing conjuncts instead of Indic text.
Dzong architecture (from Tibetan རྫོང་, Wylie rDzong, sometimes written Jong) is a distinctive type of fortress architecture found in the present and former Buddhist kingdoms of the Himalayas: Bhutan and Tibet. The architecture is massive in style with towering exterior walls surrounding a complex of courtyards, temples, administrative offices, and monks' accommodation.
Contents [hide]
1 Characteristics
2 Regional differences
2.1 Bhutan
2.2 Tibet
3 Siting of dzongs
4 Construction
5 Modern architecture in the dzong style
6 Recent scholarship
7 See also
8 References
9 Further reading
10 External links
[edit]Characteristics

Distinctive features include:
High inward sloping walls of brick and stone painted white with few or no windows in the lower sections of the wall
Use of a surrounding red ochre stripe near the top of the walls, sometimes punctuated by large gold circles.
Use of Chinese-style flared roofs atop interior temples.
Massive entry doors made of wood and iron
Interior courtyards and temples brightly colored in Buddhist-themed art motifs such as the ashtamangala or swastika, for example.
[edit]Regional differences

[edit]Bhutan
See also: History of Bhutan
Dzongs serve as the religious, military, administrative, and social centers of their district. They are often the site of an annual tsechu or religious festival.
The rooms inside the dzong are typically allocated half to administrative function (such as the office of the penlop or governor), and half to religious function, primarily the temple and housing for monks. This division between administrative and religious functions reflects the idealized duality of power between the religious and administrative branches of government.
[edit]Tibet
Further information: Administrative divisions of Tibet
The territory of Tibet used to be divided into 53 prefecture districts also called Dzongs.[1] There were two Dzongpöns for every Dzong - a lama (Tse-dung) and a layman. They were entrusted with both civil and military powers and are equal in all respects, though subordinate to the generals and the Chinese Amban in military matters,[2] until the ambans' expulsion following the fall of the Qing Dynasty in 1912.[1] Today, 71 counties in Tibet Autonomous Region are rendered as "Dzongs" in the Tibetan language.
[edit]Siting of dzongs



Trongsa Dzong, the largest dzong fortress in Bhutan.
Bhutanese dzong architecture reached its zenith in the 17th century under the leadership of the great lama Shabdrung Ngawang Namgyal. The Shabdrung relied on visions and omens to site each of the dzongs. Modern military strategists would observe that the dzongs are well-sited with regard to their function as defensive fortresses. Wangdue Phodrang dzong, for instance, is set upon a spur overlooking the confluence of the Puna Chhu and Tang Chhu rivers thus blocking any attacks by southern invaders who attempted to use a river route to bypass the trackless slopes of the middle Himalayas in attacking central Bhutan. Drukgyel dzong at the head of Paro valley guards the traditional Tibetan invasion path over the passes of the high Himalayas.


Dzong at Wangdue Phodrang, Bhutan.
Dzongs were frequently built on a hilltop or mountain spur. If the dzong is built on the side of a valley wall, a smaller dzong or watchtower is typically built directly uphill from the main dzong with the purpose of keeping the slope clear of attackers who might otherwise shoot downward into the courtyard of the main dzong below (see image at head of article). Pungtang Dechen Photrang Dzonga at Punakha is distinctive in that it is sited on a relatively flat spit of land at the confluence of the Mo Chhu and Pho Chhu rivers. The rivers surround the dzong on three sides, providing protection from attack. This siting proved inauspicious, however, when in 1994 a glacial lake 90 kilometers upstream burst through its ice dam to cause a massive flood on the Pho Chhu, damaging the dzong and taking 23 lives.
[edit]Construction



Roof construction at Tongsa dzong.
By tradition, dzongs are constructed without the use of architectural plans. Instead construction proceeds under the direction of a high lama who establishes each dimension by means of spiritual inspiration.
In previous times the dzongs were built using corvée labor which was applied as a tax against each household in the district. Under this obligation each family was to provide or hire a decreed number of workers to work for several months at a time (during quiet periods in the agricultural year) in the construction of the dzong.
Dzongs comprise heavy masonry curtain walls surrounding one or more courtyards. The main functional spaces are usually arranged in two separate areas: the administrative offices; and the religious functions - including temples and monks' accommodation. This accommodation is arranged along the inside of the outer walls and often as a separate stone tower located centrally within the courtyard, housing the main temple, that can be used as an inner defensible citadel. The main internal structures are again built with stone (or as in domestic architecture by rammed clay blocks), and whitewashed inside and out, with a broad red ochre band at the top on the outside. The larger spaces such as the temple have massive internal timber columns and beams to create galleries around an open central full height area. Smaller structures are of elaborately carved and painted timber construction.
The roofs are massively constructed in hardwood and bamboo, highly decorated at the eaves, and are constructed traditionally without the use of nails. They are open at the eaves to provide a ventilated storage area. They were traditionally finished with timber shingles weighted down with stones; but in almost all cases this has now been replaced with corrugated iron roofing. The roof of Tongsa dzong, illustrated, is one of the few shingle roofs to survive and was being restored in 2006/7.
The courtyards, usually stone-flagged, are generally at a higher level than the outside and approached by massive staircases and narrow defensible entrances with large wooden doors. All doors have thresholds to discourage the entrance of spirits. Temples are usually set at a level above the courtyard with further staircases up to them.

Pungtang Dechen Photrang Dzong and the Mo Chhu



Simtoka Dzong near Thimphu



Courtyard and tower of Rinpung Dzong at Paro

[edit]Modern architecture in the dzong style

Larger modern buildings in Bhutan often use the form and many of the external characteristics of dzong architecture in their construction, although incorporating modern techniques such as a concrete frame.
The campus architecture of the University of Texas at El Paso or UTEP is a rare example of dzong style seen outside the Himalayas. Initial phases were designed by El Paso architect Henry Trost, and later phases have continued in the same style.[3]

To the left is the College of Business, to the right the College of Engineering



UTEP's Academic Services Building



UTEP Library

[edit]Recent scholarship

Recent research by the prominent Bhutanese scholar C.T. Dorji suggests that the original 'model dzong' may not be Simtokha Dzong as commonly believed, but Dobji Dzong, built in 1531 at an altitude of 6600 feet on a cliff facing the gorge of the Wong Chhu. Unlike the dzongs built under the direction of the Shabdrung for defensive purposes, Dobji Dzong was constructed to serve a religious purpose, marking the spot where Ngawang Chogyel "...followed a spring water which originated from beneath the throne of Jetsun Milarepa in Druk Ralung to find a suitable site for establishing a center to propagate Drukpa Kagyu teachings in Bhutan".[4]
[edit]See also

Architectural style
Architecture of Bhutan
Driglam Namzha
[edit]References

^ a b Le Tibet, Marc Moniez, Christian Deweirdt, Monique Masse, Éditions de l'Adret, Paris, 1999, ISBN 2-907629-46-8
^ Das, Sarat Chandra. (1902). Lhasa and Central Tibet. Reprint (1988): Mehra Offset Press, Delhi, p. 176.
^ For more details see the UTEP Handbook of Operations.
^ [1][dead link]
[edit]Further reading

Amundsen, Ingun B (Winter 2001). "On Bhutanese and Tibetan Dzongs" (PDF). Journal of Bhutan Studies 5: 8–41.
Bernier, Ronald M. (1997). Himalayan Architecture. Fairleigh Dickinson University Press. ISBN 0-8386-3602-0.
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Dzongs of Bhutan
Designs for Trashi Chhoe Dzong Precinct, Thimphu, Bhutan
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