Symmetry

Symmetry (from Greek συμμετρεῖν symmetría "measure together") generally conveys two primary meanings. The first is an imprecise sense of harmonious or aesthetically pleasing proportionality and balance; such that it reflects beauty or perfection. The second meaning is a precise and well-defined concept of balance or "patterned self-similarity" that can be demonstrated or proved according to the rules of a formal system: by geometry, through physics or otherwise.

Although the meanings are distinguishable in some contexts, both meanings of "symmetry" are related and discussed in parallel.

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.