Symmetry (from Greek συμμετρία symmetria ”agreement in dimensions, due proportion, arrangement”) 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.