The Art and Science of Symmetry

SYMMETRY
Symmetry is a characteristic feature of geometrical shapes, systems, equations, and other real or conceptual objects —typically, in which one half of the object appears to be a reflection (i.e., a "mirror") of the other half.

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).

Similarity vs. Sameness
Although two objects with great similarity appear the same, they must logically be different. For example, if one rotates an equilateral triangle around its 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 indistinguishable from its previous form. In reality however, each corner of any equilateral triangle composed of matter must be composed of separate molecules in separate locations. Therefore, symmetry in real physical objects is a matter of similarity instead of sameness. The difficulty for an intelligence to differentiate such a seemingly exact similarity is understandable.

Symmetry in Aesthetics
Symmetry does not by itself confer beauty to an object — many symmetrical designs can be boring or overly challenging, and on the other hand preference for, or dislike of, exact symmetry is apparently dependent on cultural background. Along with texture, color, proportion, and other factors, symmetry does however play an important role in determining the aesthetic appeal of an object