Symmetry 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
* 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 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.
The most familiar type of symmetry for many people is geometrical symmetry. A geometric object is said to be symmetric if, after it has been geometrically transformed, it retains some property of the original object (i.e., the object has an invariance under the transform). 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. 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. Reflection symmetry
Reflection 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...
Please join StudyMode to read the full document