symmetrical group o.]] 's Vitruvian Man (ca. 1487) is often used as a representation of symmetry in the human body and, by extension, the natural universe.]]

Symmetry (from the Greek: "συμμετρεῖν" = to measure together), generally conveys two primary meanings. The first is an imprecise sense of harmonious or aesthetically pleasing proportionality and balance;

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; and as an aspect of abstract objects, theoretic models, language, music and even knowledge itself.

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.

Symmetry 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.

Reflection 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. Note that this is sometimes called horizontal symmetry, and sometimes 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.

In certain contexts there is rotational symmetry anyway. Then mirror-image symmetry is equivalent with inversion symmetry; in such contexts in modern physics the term P-symmetry is used for both (P stands for parity).

For more general types of reflection there are corresponding more general types of reflection symmetry. Examples:

with respect to a non-isometric affine involution (an oblique reflection in a line, plane, etc.).

with respect to circle inversion

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.

In another meaning of the word, the rotation group of an object is the symmetry group within E⁺(n), 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

Translational symmetry leaves an object invariant under a discrete or continuous group of translations $T\_a(p)=p+a$.

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.

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 S_2n of order 2n (not to be confused with symmetric groups, for which the same notation is used; abstract group C_2n); 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.

C_nh (angle of 360°/n); for odd n this is generated by a single symmetry, and the abstract group is C_2n, for even n this is not a basic symmetry but a combination. See also point groups in three dimensions.

Helical symmetry

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 $\backslash theta$ 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°/$\backslash theta$, see e.g. double helix. This concept can be further generalized to include cases where $m\backslash theta$ 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 $\backslash theta$ 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.

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 dilations are 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.

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, 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.

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).

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.

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.

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" (∧, $\backslash land$, or &), "or" (∨), "biconditional" (if and only if) (↔), NAND ("not-and"), XOR ("not-biconditional"), and NOR ("not-or").

Symmetry in science

Symmetry in physics

Symmetry in physics has been generalized to mean invariance—that is, lack of any visible 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.

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

Symmetry in biology

Symmetry in chemistry

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

Symmetry 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.

Symmetry in religious symbols

[[File:Symmetric religious symbols.svg|thumb|Symmetry in religious symbols. Row 1. Christian, Jewish, Taoism Row 2. Islamic, Buddhist, Shinto Row 3. Sikh, Baha'i, Jain ]] The tendency of people to see purpose in symmetry suggests at least one reason why symmetries are often an integral part of the symbols of world religions. Just a few of many examples include the sixfold rotational symmetry of Judaism's Star of David, the twofold point symmetry of Taoism's Taijitu or Yin-Yang, the bilateral symmetry of Christianity's cross and Sikhism's Khanda, or the fourfold point symmetry of Jain's ancient (and peacefully intended) version of the swastika. With its strong prohibitions against the use of representational images, Islam, and in particular the Sunni branch of Islam, has developed intricate use of symmetries.

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.

Symmetry in architecture

Another human endeavor in which the visual result plays a vital part in the overall result is architecture. Both in ancient 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 examples of 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.

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 caved doors, stained glass windows, tile mosaics, friezes, stairwells, stair rails, and balustradess. 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.

Symmetry 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. 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 B.C. 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 B.C. Bronze vessels exhibited both a bilateral main motif and a repetitive translated border design.

Symmetry in quilts

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.

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.

Symmetry in music

File:Major and minor triads.png|300px|thumb|right|Major and minor triads on the white piano keys are symmetrical to the D. (compare article) (file)

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desc none Symmetry is of course 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) 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).

{| |rowspan=3|+ |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)

Equivalency

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

Symmetry in other arts and crafts

work]] 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.

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. 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. 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. 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. Most people display a preference for figures that have a certain degree of simplicity and symmetry, but enough complexity to make them interesting.

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.

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).

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; 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.

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 & 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

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

References

External links

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

*

Name	Philip Oakey
Landscape	Yes
Background	solo_singer
Alias	Phil Oakey
Born	October 02, 1955
Origin	Hinckley, Leicestershire, England
Instrument	Vocals, Keyboards
Voice type	Baritone
Genre	Synthpop, Electronica
Occupation	Singer, Composer, Producer
Years active	1977–present
Label	Fast Product, Virgin Records, EMI, EastWest, Papillon
Associated acts	The Human LeagueGiorgio MoroderRespectAll Seeing IKings Have Long ArmsHiemPet Shop BoysLittle Boots

Philip Oakey (born on 2 October 1955, in Hinckley, Leicestershire, England), is a composer, singer, songwriter and producer.

He is best known as the lead singer, frontman and co-founder of the famous English synthpop band The Human League. He has also had an extensive solo music career and collaborated with numerous other artists and producers. He is also an occasional DJ.

Early life

Oakey was born on 2 October 1955 in Hinckley, Leicestershire, but moved to Sheffield at an early age when his father changed jobs to work in the city. He was educated at King Edward VII School in Sheffield. He left school at 18 without finishing his exams and worked in a number of casual jobs: in a university bookshop, and from 1975 as a porter at Thornbury Annex Hospital in Sheffield. He was married briefly to his girlfriend, whom he met at school, but the marriage did not last long and they were divorced in 1980.

Entry into music

Oakey’s entry into music in 1977 was entirely accidental. He had bought a saxophone but had given up trying to learn how to play it, and had no aspiration to be in a pop group. In Sheffield in 1977 Martyn Ware (a school friend of Oakey’s), Ian Craig Marsh and Adi Newton

Human League career

In late 1977 The Future changed its name to ‘The Human League’ - after an element of a science fiction board game. The new band played their first live gig at Psalter Lane Arts College in June 1978 (a blue plaque now marks the spot) and signed to Fast Records. The early Human League had a reputation for being arty and had only limited commercial success, releasing two singles, "Being Boiled" and "Empire State Human", with lyrics written by Oakey. They would eventually release two albums, Reproduction (1979), and Travelogue (1980), both recorded at the band's Monumental Pictures studio, but dogged by the lack of commercial success Oakey and Ware’s working relationship became increasingly strained. In October 1980, on the eve of a European tour, it reached breaking point and Ware walked out taking Marsh with him. Oakey and director of visuals Adrian Wright were permitted to retain the band name but would be responsible for all band debts and the tour commitment. Ware and Marsh soon recruited Glenn Gregory and became Heaven 17. With the tour promoters threatening to sue Oakey and facing financial ruin, he had less than a week to put a new band together. In an unplanned move that is now entrenched in popular folklore, Oakey went to a Sheffield city centre discothèque called The Crazy Daisy (since demolished) and recruited two totally unknown teenage schoolgirls: Susan Ann Sulley and Joanne Catherall into the band. As luck would have it, the women were already fans of The Human League and recognised Oakey. He now calls this the best decision of his career, as the girls would be critical in the band's further success, and Sulley and Catherall are now Oakey's business partners in the present day band.

The remainder of the 1980s saw the band's success peak and dip, with the follow-up release of the album Hysteria in 1984 underachieving. In 1986 Oakey accepted an offer to work with U.S. producers Jimmy Jam and Terry Lewis which resulted in the release of the album Crash and the single "Human" which became another international hit and went to number one in the U.S. By 1987 the band had lost most of its original members leaving only Oakey, Sulley and Catherall. In 1989 Oakey persuaded Sheffield City Council to invest in business development loan for the building of Human League Studios in Sheffield, Oakey's dedicated studio for the band and a commercial venture. The 1990 album Romantic? failed to sell and in 1992 Virgin records cancelled the band's recording contract. This had a devastating effect on the band, causing Oakey to seek counselling for depression, and Sulley to have a breakdown. The emotional problems of the pair nearly caused the band to fold. Thanks mainly to the efforts of Catherall, by 1993 Oakey and Sulley had recovered and the band was back on its feet. They signed to Eastwest records which resulted in the release of the gold selling album Octopus and the hit singles "Tell Me When" and "One Man in My Heart".

Another change of record label saw the release of critically acclaimed Secrets album in 2001. Secrets failed to sell because the record label went into receivership curtailing promotion. After the failure of a project he had put so much work and time into, Oakey lost faith in the record industry and changed the band's focus in to more lucrative live work. Between 2002 and the present day they have toured almost constantly, either on their own or as guests at festivals. They have played at such prestigious events as V festival, Festival Internacional de Benicàssim and to 18,000 fans at the Hollywood Bowl, Los Angeles in 2006.

Solo and collaborative career

Oakey's career in music has rarely been confined to just the Human League. He has worked on his own, but also with numerous other artists and producers. His first collaboration was producing the Spanish released single "Amor Secreto" by Nick Fury in 1983 for which he also played synthesizer, together with Jo Callis.

In 1985 Oakey and Moroder would release the joint album Philip Oakey & Giorgio Moroder which would generate two single releases, "Be My Lover Now" and "Good-Bye Bad Times". Released in both the UK and U.S. these singles were not as successful as "Together in Electric Dreams" and the Oakey/Moroder partnership was effectively ended.

In 1990 Oakey guest vocalled on "What Comes After Goodbye", the one-off release by the short lived Sheffield dance band 'Respect'. In 1999 he provided vocals for the single "1st Man in Space" by another Sheffield band All Seeing I. The song was written by Jarvis Cocker.

In 1991 Oakey worked as Producer on comedian, Vic Reeves only studio album, I Will Cure You - most notably on the track "Black Night" which is a Deep Purple cover.

In 2003 he provided vocals for another Sheffield band Kings Have Long Arms on the single "Rock and Roll is Dead"; also in 2003 he worked with producer/DJ Alex Gold and they released the trance single "LA Today". In 2008 Oakey worked with Hiem, a band fronted by former All Seeing I lead singer David "Bozz" Boswell, on the song "2 am".

Oakey (often with Neil Sutton) has written music for third parties which has been released anonymously or published uncredited. He has infrequently guest DJed, and one of his sets entitled 'the history of the synthesizer' was streamed live over the internet.

In early 2009 Oakey collaborated with Pet Shop Boys on their tenth studio album Yes, supplying vocals for the intended bonus disc song "This Used to Be the Future". Also in 2009 Oakey collaborated with British female synthpop artist Little Boots on her first album Hands, recording the duet track "Symmetry".

Pre-1977, during the era of Punk rock, Oakey adopted various styles; at one time having a crew cut but later he had collar length hair and had once turned up in one club wearing a household power lead with plug as a necklace. He also often wore bike leathers and rode a distinctive Old Norton motorcycle around Sheffield. His natural good looks combined with his flamboyant style was the main reason Martyn Ware invited him to join his pop band ‘The Future’ in 1977. Ware, who was chasing commercial success, reasoned that half the battle was won "as Oakey already looked like a pop star".

Soon after 'The Future' transformed into 'The Human League', Oakey as lead singer wanted a look that would make him stand out from other lead singers. After spotting a girl on a Sheffield bus with a Veronica Lake hair style, Oakey was inspired to adopt a strange lopsided geometric hairstyle, shoulder length on one side and short on the other. As the Human League increased in prominence the hairstyle would became Oakey’s trademark. Between 1978-1979 with his unique hairstyle he maintained a masculine dress style and at one time wore a full beard.

Increasingly interested in attracting the limelight and standing out from the crowd, in 1979 inspired by the 1970s Glamrock style of Brian Eno Oakey began wearing makeup; his style became increasingly more feminine including the use of bright red lipstick.

By 1981 after the formation of the Mk2 Human League, Oakey’s trademark style of the early 1980s was complete. As well as full make up, Oakey had begun wearing androgynous clothing, which he described as "neither male nor female". The addition of teenage school girls Susan Ann Sulley and Joanne Catherall as co-vocalists to the band in 1980 complemented his look. At times all three would wear the same eyeliner and lipstick. Oakey and Catherall, who were to enter into a relationship with each other, often looked and dressed almost identically.

The media regularly commented and joked about his style (unwittingly achieving Oakey's aim). Fearlessly Oakey pushed his unique style further and began wearing high heeled shoes. He already had both his ears pierced and wore dangling women’s diamante earrings. Keen to shock, on one of the 'new' Human League's posters in 1981 Oakey posed shirtless displaying pierced nipples linked together by a gold chain. Oakey says of his early 1980s style: "I deliberately wore clothes that either men or women could wear. But I don't think I ever really looked like a woman. And I never wore very masculine clothes". Oakey jokes that when he sought parental permission for the girls to go on the 1980 tour, that the father of Susan Sulley (then aged 17) only let her go on tour with the Human League because “He wasn’t entirely sure I was a man”.

After the international success of Dare, Oakey tired of the androgynous look and then by 1983 had adopted a more macho look of denim, collar length permed hair and the ill shaven ‘designer stubble’.

For the Crash album of 1986 Oakey adopted a smoother style of designer clothes of the period and a very manicured look which he says was inspired by Sean Young’s character ‘Rachael’ from the film Blade Runner.

By 1990 the Human League had seriously begun to decline. For their Romantic? album, Oakey wore denim, leather and readopted his lop sided hairstyle from 1981 in a rebellion against "the male model look of Crash". The band went through dark times and the style was quickly abandoned.

When the band returned in a comeback in 1995, the mature (then 40 year old) Oakey reappeared with designer clothes and a sauvé short neat hair cut.

In 1998 Oakey began to suffer from male pattern baldness and after advice from his hair stylist, in 1999 he adopted an all over ‘number two’ crop hairstyle. This is the style he wears today, albeit that his hair has now completely greyed.

Today Oakey is still known for his dapper style, but now generally wears a simple Armani suit on stage. Although he has not lost his desire to shock, and recently boasted during a newspaper interview that he had recently acquired a Prince Albert piercing, which he says “hurt less than having his ears pierced”.

Today

Today Oakey still lives in central Sheffield with his long term girlfriend, close to his studio; and long term colleagues Sulley and Catherall. He continues to work full time in the music industry, principally touring with The Human League. When not touring he works in the studio or with other artists. He has stated that he is currently working on a solo studio album, although he refuses to be drawn on a timescale for its release.

Discography

with The Human League

;Studio albums

Reproduction (1979)

Travelogue (1980)

Dare (1981)

Love and Dancing (1982)

Hysteria (1984)

Crash (1986)

Romantic? (1990)

Octopus (1995)

Secrets (2001)

;Number-one singles

"Don't You Want Me" - 1981 (UK)(25th highest selling single of all time), 1982 (U.S, AUS, NZ)

"Human" - 1986 (U.S)

Solo

;Albums

"Philip Oakey & Giorgio Moroder'" Virgin, 1985

;Singles

"Together In Electric Dreams" - (with Giorgio Moroder), Virgin Records, 1984

"Good-Bye Bad Times" - (with Giorgio Moroder), Virgin, 1985

"Be My Lover Now" - (with Giorgio Moroder), Virgin, 1985

"What Comes After Goodbye" - (Respect featuring Philip Oakey), Chrysalis Records, 1990

"1st Man in Space" - (All Seeing I, vocals by Philip Oakey), FFRR Records, 1999

"Rock and Roll is Dead" - (Kings Have Long Arms featuring Philip Oakey), Heart and Soul Records, 2003

"LA Today" - Alex Gold (featuring Philip Oakey), Xtravaganza Records, 2003

Film and television

1990 "The Weekenders" (TV) (D. Vic Reeves) - Played Himself

1999 "Hunting Venus" (Buffalo Films, D. Martin Clunes) - Played Himself

2010 "Top Gear 15x01" - Guest Appearance (Jeremy Clarkson's Three-Wheeler Report)

Awards

1982 BRIT Awards - (as 'The Human League') - 'Best British Breakthrough Act'

2004 Q Awards - (as 'The Human League') - 'The Q Innovation in Sound Award'

Nominated for Grammy Award in 1982 for Best International Act (as 'The Human League')

Further reading

Story of a Band Called "The Human League" by Alaska Ross (Proteus July 1982) ISBN 978-0862761035

See also

External links

Oakey deliberately does not have an official website, not wanting to do what others do, and apparently believing it is expensive to have one.

References

Category:English male singers Category:1955 births Category:Living people Category:Old Edwardians (Sheffield) Category:People from Hinckley Category:People from Sheffield Category:The Human League members

Name	Little Boots
Background	solo_singer
Birth name	Victoria Christina Hesketh
Born	April 05, 1984 also known by her stage name/pseudonym Little Boots, is an English electropop singer-songwriter.
Name	Little Boots
Alternative names	Hesketh, Victoria
Short description	Electronic musician and singer
Date of birth	5 April 1984
Place of birth	Blackpool, Lancashire, England

Category:1984 births Category:679 Artists artists Category:Alumni of the University of Leeds Category:Atlantic Records artists Category:Elektra Records artists Category:English electronic musicians Category:English female singers Category:English pop pianists Category:English pop singers Category:English singer-songwriters Category:Living people Category:Old Rossallians Category:People from Blackpool Category:People from Thornton-Cleveleys Category:Synthpop Category:Wonky Pop acts