Symmetry, Symmetry

The second 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 third perspective discusses symmetry in the humanities, covering its rich and varied use in history, architecture, art, and religion.

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.

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.

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.

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.

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.

For example, in 1D, the symmetry group of 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.

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).

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.

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.

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).

Rotations are direct isometries, i.e., isometries preserving orientation. Therefore a symmetry group of rotational symmetry is a subgroup of E +.

Because of Noether's theorem, rotational symmetry of a physical system is equivalent to the angular momentum conservation law. See also rotational invariance.

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.

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.

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.

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 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.

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 and visually impressive use of symmetries.

The image, which is often misunderstood in the Western world as representing good (white) versus evil (black), is actually intended as a graphical representative of the complementary need for two abstract concepts of "maleness" (white) and "femaleness" (black). The symmetry of the symbol in this case is used not just to create a symbol that catches the attention of the eye, but to make a significant statement about the philosophical beliefs of the people and groups that use it.

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.

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, Bla Bartk, and George Perle have used axes of symmetry and/or interval cycles in an analogous way to keys or non-tonal tonal center.

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.

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.

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.

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.

Source: Wikipedia > Symmetry