Symmetry 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.See e.g.,

This article describes these notions of symmetry from three perspectives. The first is that of mathematics, in which symmetries are defined and categorized precisely. 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.

The opposite of symmetry is asymmetry.

Symmetry in the field of mathematics

In formal terms, we say that an object is symmetric with respect to a given mathematical operation, if, when applied to the object, this operation does not change the object or its appearance. Two objects are symmetric to each other with respect to a given group of operations if one is obtained from the other by some of the operations (and vice versa).

Symmetries may also be found in living organisms including humans and other animals (see symmetry in biology below). In 2D geometry the main kinds of symmetry of interest are with respect to the basic Euclidean plane isometries: translations, rotations, reflections, and glide reflections.

Mathematical model for symmetry

The set of all symmetry operations considered, on all objects in a set X, can be modeled as a group action g : G × X → X, where the image of g in G and x in X is written as g·x. 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. G is the symmetry group of the space itself, and of any object that is uniform throughout space. 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.