David W. Farmer in American Scientist:
Symmetry is a fundamental concept pervading both science and culture. In popular terms, symmetry is often viewed as a kind of “balance,” as when Doris Day’s character in the 1951 movie On Moonlight Bay insists that if her beau kisses her on the right cheek, then he should kiss her on the left cheek too. But in mathematics, symmetry has been given a more precise meaning. In his new history of mathematical symmetry, Why Beauty Is Truth, Ian Stewart gives this definition: “A symmetry of some mathematical object is a transformation that preserves the object’s structure.” So a symmetrical structure looks the same before and after you do something to it. A butterfly looks the same as its mirror image. The (idealized) wheel of a car may look the same after being rotated on its axle by 90 degrees (or possibly by 72 or 120 degrees, depending on the particular design).
Although mathematical symmetry may bring to mind a regular polygon or other geometric pattern, its roots (pun unavoidable) lie in algebra, in the solutions to polynomial equations. Thus Stewart begins his account in ancient Babylon with the solution to quadratic equations. The familiar quadratic formula gives the two roots of the degree-two polynomial equation ax 2 + bx + c = 0. The Babylonians didn’t have the algebraic notation to write down such a formula, but they had a recipe that was equivalent to it.