Symmetry

Symmetry is a characteristic of geometrical shapes, equations and other objects; we say that such an object is symmetric with respect to a given operation if this operation, when applied to the object, does not appear to change it. The three main symmetrical operations are reflection, rotation and translation. A reflection "flips" an object over a line, inverting it as if in a mirror. A rotation rotates an object using a point as it's center. A translation (geometry) "slides" an object from one area to another by a vector. Even more complex operations on a geometric object, like shrinking or shape warping, can be reduced to the operation of translation of every point within the object. Symmetry occurs in geometry, mathematics, physics, biology, art, literature (palindromes), etc.

Although two objects with great similarity appear the same, they must logically be different. For example, if one rotates an equilateral triangle around it's center 120 degrees, it will appear the same as it was before the rotation to an observer. In theoretical euclidean geometry, such a rotation would be unrecognizable from it's previous form. In reality however, each corner of any equilateral triangle composed of matter must be composed of separate molecules in separate locations. Symmetry therefore, is a matter of similarity instead of sameness. The difficulty for an intelligence to differentiate such a seemingly exact similarity might be responsible for the mild altered state of consciousness one gets by observing intricate patterns based on symmetry.

Table of contents

1 Symmetry in geometry 2 Symmetry in mathematics 2.1 Generalization of symmetry 3 Symmetry in physics 4 Symmetry in biology 5 Symmetry in art 6 Symmetry in literature

Symmetry in geometry

The object with the most symmetry is empty space because any part of can be rotated, reflected or translated without apparent change.

The most familiar and conventionally taught type of symmetry is the left-right or mirror image symmetry exhibited for instance by the letter T: when this letter is reflected along a vertical axis, it appears the same. An equilateral triangle exhibits such a reflection symmetry along three axes, and in addition it shows rotational symmetry: if rotated by 120 or 240 degrees, it remains unchanged. An instance of a shape which exhibits only rotational but no reflectional symmetry is the swastika.

The German geometer Felix Klein enunciated a very influential Erlangen programme in 1872, suggesting symmetry as unifying and organising principle in geometry (at a time when that was read 'geometries'). This is a broad rather than deep principle. Initially it led to interest in the groups attached to geometries, and the slogan transformation geometry (an aspect of the New Math, but hardly controversial in modern mathematical practice). By now it has been applied in numerous forms, as kind of standard attack on problems.

A fractal, coined by Mandelbrot is symmetry involving scale. For example an equilateral triangle can be shrunk so that each of it's sides are one third the length of the original's sides. These smaller triangles can be rotated and translated until they are adjacent and in the center of each of the larger triangle's lines. The smaller triangles can repeat the process, resulting in even smaller triangles on their sides. Fascinating intricate structures can be created by repeating such scaling symmetrical operations many times.

Symmetry in mathematics

An example of a mathematical expression exhibiting symmetry is a²c + 3ab + b²c. If a and b are exchanged, the expression remains unchanged due to the commutativity of addition and multiplication.

In mathematics, one studies the symmetry of a given object by collecting all the operations that leave the object unchanged. These operations form a group. For a geometrical object, this is known as its symmetry group; for an algebraic object, one uses the term automorphism group. The whole subject of Galois theory deals with well-hidden symmetries of fields.

Generalization 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.

Symmetry in physics

The generalisation of symmetry in physics to mean invariance under any kind of transformation has become one of the most powerful tools of theoretical physics. See Noether's theorem for more details. This has led to group theory being one of the areas of mathematics most studied by physicists.

Symmetry in biology

See Bilateral symmetry...

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Symmetry in art

See M. C. Escher

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Symmetry in literature

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See also: chirality, Bilateral symmetry