# Symmetry

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Symmetry (from Greek συμμετρία symmetria "agreement in dimensions, due proportion, arrangement") has two meanings. The first is a vague sense of harmonious and beautiful proportion and balance. The second is an exact mathematical "patterned self-similarity" that can be demonstrated with the rules of a formal system, such as geometry or physics.

Although these two meanings of "symmetry" can sometimes be told apart, they are related, so they are here discussed together.

Mathematical symmetry may be observed

This article describes these notions of symmetry from four perspectives. The first is symmetry in geometry, which is the most familiar type of symmetry for many people. The second is the more general meaning of symmetry in mathematics as a whole. The third 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. The fourth discusses symmetry in the humanities, covering its rich and varied use in history, architecture, art, and religion.

The opposite of symmetry is asymmetry.

## Geometry

The most familiar type of symmetry for many people is geometrical symmetry. A geometric figure (object) has symmetry if there is an isometry that maps the figure onto itself (i.e., the object has an invariance under the transform). If the isometry is the reflection of a plane figure, the figure has reflectional symmetry or line symmetry. For instance, a circle rotated about its center will have the same shape and size as the original circle. A circle is then said to be symmetric under rotation or to have rotational symmetry. It is possible for a figure to have more than one line of symmetry.

The type of symmetries that are possible for a geometric object depend on the set of geometric transforms available and what object properties should remain unchanged after a transform. Because the composition of two transforms is also a transform and every transform has an inverse transform that undoes it, the set of transforms under which an object is symmetric form a mathematical group.

The most common group of transforms considered is the Euclidean group of isometries, or distance preserving transformations, in two-dimensional (plane geometry)or three-dimensional (solid geometry) Euclidean space. These isometries consist of reflections, rotations, translations and combinations of these basic operations. Under an isometric transformation, a geometric object is symmetric if the transformed object is congruent to the original.

A geometric object is typically symmetric only under a subgroup of isometries. The kinds of isometry subgroups are described below, followed by other kinds of transform groups and object invariance types used in geometry.

### Reflectional symmetry

{{#invoke:main|main}} An isosceles triangle with mirror symmetry. The dashed line is the axis of symmetry. Folding the triangle across the axis results in two identical halves.

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

In one dimension, there is a point of symmetry about which reflection takes place; in two dimensions there is an axis of symmetry, and in three dimensions there is 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 passing through its center, 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 reflection 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 isomorphic to C2. The fundamental domain is a half-plane or half-space.

### Point reflection and other involutive isometries

{{#invoke:main|main}} Two triangles showing point reflection symmetry in the plane. Triangle A'B'C' can also be generated from triangle ABC by a 180° rotation around point O.

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

(x1, … xm) ↦ (−x1, … −xk, xk+1, … xm)

in a certain system of Cartesian coordinates. This reflects the space along an mk-dimensional affine subspace. If Template:Mvar = Template:Mvar, then such a transformation is known as a point reflection or an inversion through a point. On the plane (Template:Mvar = 2) a point reflection is the same as a half-turn (180°) rotation; see below. Antipodal symmetry is an alternative name for a point reflection symmetry through the origin.

Such a "reflection" preserves orientation if and only if Template:Mvar is an even number. This implies that for [[three-dimensional space|Template:Mvar = 3]] (as well as for other odd Template:Mvar) a point reflection changes the orientation of the space, like a mirror-image symmetry. That is why in physics the term P-symmetry is used for both point reflection and mirror symmetry (P stands for parity). As a point reflection in three dimensions changes a left-handed coordinate system into a right-handed coordinate system, symmetry under a point reflection is also called a left-right symmetry.

### Rotational symmetry

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Rotational symmetry is symmetry with respect to some or all rotations in Template:Mvar-dimensional Euclidean space. Rotations are direct isometries; i.e., isometries preserving orientation. Therefore a symmetry group of rotational symmetry is a subgroup of the special Euclidean group [[SE(3)|E+(Template:Mvar)]].

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+(Template:Mvar). 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(Template:Mvar), which can be represented by the group of m × m orthogonal matrices with determinant 1. For Template:Mvar = 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+(Template:Mvar), 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.

### Translational symmetry

{{#invoke:main|main}} Translational symmetry leaves an object invariant under a discrete or continuous group of translations $T_{a}(p)\;=\;p\,+\,a$ . The illustration on the right shows four congruent triangles generated by translations along the arrow. If the line of triangles extended to infinity in both directions, they would have a discrete translational symmetry; any translation that mapped one triangle onto another would leave the whole line unchanged.

### Glide reflection symmetry A glide reflection in which the upper triangle is reflected about the horizontal axis of symmetry and then translated to the right.

In the plane, a glide reflection symmetry (in 3D it is called 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. The composition of two glide reflections results in a translation symmetry with twice the translation vector. The symmetry group comprising glide reflections and associated translations is the frieze group p11g and is isomorphic with the infinite cyclic group Z.

### Rotoreflection symmetry A rotoreflection in 3D in which the upper triangle is rotated about a vertical axis and then reflected about a horizontal plane of symmetry.

In 3D, a rotoreflection or improper rotation is a rotation about an axis combined with reflection in a plane perpendicular to that axis. The symmetry groups associated with rotoreflections include:

• if the rotation angle has no common divisor with 360°, the symmetry group is not discrete
• if the rotoreflection has a 2n-fold rotation angle (angle of 180°/n), the symmetry group is S2n of order 2n (not to be confused with symmetric groups, for which the same notation is used; the abstract group is C2n). A special case is n = 1, an inversion, because it does not depend on the axis and the plane, it is characterized by just the point of inversion.
• the group 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.

### Helical symmetry

File:400px-Drillbit.jpg
A drill bit with helical symmetry.

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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 constant angular speed while simultaneously translating at a constant linear speed along its axis of rotation. At any one point in time, these two motions combine to give a coiling angle that helps define the properties of the traced helix. 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°; for example, a double helix. This concept can be further generalized to include cases where $m\theta$ 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, with approximately 10.5 base pairs per turn, is an example of this type of non-repeating helical symmetry.

### 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.

### 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 (so-called allometric scaling), 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.

## In mathematics

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Generalizing from geometrical symmetry in the previous section, 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).

### 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 × XX, 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: VV acting on the set of functions x: VW 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(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 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.

### Symmetric functions

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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 x2yz 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.

### 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 ⊽).

## In science and nature

### In physics

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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.

### In physical objects

#### 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.

#### 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.

#### 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.

### 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 $G\;\to \;\mathrm {Aut} (G)$ , 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.

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

### In biology

Bilateral animals, including humans, are more or less symmetric with respect to the sagittal plane which divides the body into left and right halves. Animals that move in one direction necessarily have upper and lower sides, head and tail ends, and therefore a left and a right. The head becomes specialized with a mouth and sense organs, and the body becomes bilaterally symmetric for the purpose of movement, with symmetrical pairs of muscles and skeletal elements, though internal organs often remain asymmetric.

Plants and sessile (attached) animals such as sea anemones often have radial or rotational symmetry, which suits them because food or threats may arrive from any direction. Fivefold symmetry is found in the echinoderms, the group that includes starfish, sea urchins, and sea lilies.

### In chemistry

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

## In history, religion, and culture

### 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.

### In architecture

Architects have long employed overall symmetry as a method of telling the viewer that a building is intended to convey an important idea, that the building is a physical expression of an important belief and ideals of the people who commissioned it. Symmetry endows a formal quality that distinguishes itself from the asymmetrical forms that are more casual. Such a building with an expressive intention is therefore, by definition, monumental, as it seeks to memorialize an idea. Symmetry is an expressive device, often used in conjunction with other methods like magnified size, elevation above ground level, and use of permanent materials. All are ways of communicating the importance and validity of the idea that is intended to be memorialized.

Examples of ancient architectures that made powerful use of symmetry to convey importance 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.

An example of a broken symmetry in architecture is the Leaning Tower of Pisa, whose notoriety stems not from the intended symmetry of its design, but from the violation of that symmetry from the lean that developed while it was still under construction. It conveys the failure of the idea it sought to memorialize. 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.Modernist architecture, starting with International style, sometimes rejects symmetry, relying on wings and balance of masses.{{ safesubst:#invoke:Unsubst||date=__DATE__ |$B= {{#invoke:Category handler|main}}{{#invoke:Category handler|main}}[citation needed] }} ### In pottery and metal vessels Since the earliest uses of pottery wheels to help shape clay vessels, pottery has had a strong relationship to symmetry. Pottery created using a wheel acquires full rotational symmetry in its cross-section, while allowing substantial freedom of shape in the vertical direction. Upon this inherently symmetrical starting point, potters from ancient times onwards have added patterns that modify the rotational symmetry to achieve visual objectives. For example, Persian pottery dating from the fourth millennium BC and earlier used symmetric zigzags, squares, cross-hatchings, and repetitions of figures to produce visually striking overall designs.{{ safesubst:#invoke:Unsubst||date=__DATE__ |$B= {{#invoke:Category handler|main}}{{#invoke:Category handler|main}}[citation needed] }}

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.

### In quilts

File:Kitchen kaleid.svg
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.

### In carpets and rugs

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.

### In music

<imagemap> File:Major and minor triads.png|thumb|right|Major and minor triads on the white piano keys are symmetrical to the D. (compare article) (file)

poly 35 442 35 544 179 493 root of A minor triad poly 479 462 446 493 479 526 513 492 third of A minor triad poly 841 472 782 493 840 514 821 494 fifth of A minor triad poly 926 442 875 460 906 493 873 525 926 545 fifth of A minor triad poly 417 442 417 544 468 525 437 493 469 459 root of C major triad poly 502 472 522 493 502 514 560 493 root of C major triad poly 863 462 830 493 863 525 895 493 third of C major triad poly 1303 442 1160 493 1304 544 fifth of C major triad poly 280 406 264 413 282 419 275 413 fifth of E minor triad poly 308 397 293 403 301 412 294 423 309 428 fifth of E minor triad poly 844 397 844 428 886 413 root of E minor triad poly 1240 404 1230 412 1239 422 1250 412 third of E minor triad poly 289 404 279 413 288 422 300 413 third of G major triad poly 689 398 646 413 689 429 fifth of G major triad poly 1221 397 1222 429 1237 423 1228 414 1237 403 root of G major triad poly 1249 406 1254 413 1249 418 1265 413 root of G major triad poly 89 567 73 573 90 579 86 573 fifth of D minor triad poly 117 558 102 563 111 572 102 583 118 589 fifth of D minor triad poly 650 558 650 589 693 573 root of D minor triad poly 1050 563 1040 574 1050 582 1061 574 third of D minor triad poly 98 565 88 573 98 583 110 574 third of F major triad poly 498 558 455 573 498 589 fifth of F major triad poly 1031 557 1031 589 1047 583 1038 574 1046 563 root of F major triad poly 1075 573 1059 580 1064 573 1058 567 root of F major triad

desc none </imagemap>

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.

#### 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.

#### 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)Template:Clarify 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).

#### Equivalency

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

### In other arts and crafts

Symmetries appear in the design of objects of all kinds. Examples include beadwork, furniture, sand paintings, knotwork, masks, and musical instruments. Symmetries are central to the art of M.C. Escher and the many applications of tiling.

### In aesthetics

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The relationship of symmetry to aesthetics is complex. Humans use bilateral symmetry to judge the likely health or fitness of other people.{{ safesubst:#invoke:Unsubst||date=__DATE__ |\$B= {{#invoke:Category handler|main}}{{#invoke:Category handler|main}}[citation needed] }} Opposed to this is the tendency for excessive symmetry to be perceived as boring or uninteresting. People prefer shapes that have some symmetry, but enough complexity to make them interesting.

## See also

Template:Colbegin Symmetry in statistics

• Skewness, asymmetry of a statistical distribution

Symmetry in games and puzzles

Symmetry in literature

Moral symmetry

Other