The Symmetries of Things

Start with a single shape. Repeat it in some way—translation, reflection over a line, rotation around a point—and you have created symmetry.Symmetry is a fundamental phenomenon in art, science, and nature that has been captured, described, and analyzed using mathematical concepts for a long time. Inspired by the geometric intuition of Bill Thurston and empowered by his own analytical skills, John Conway, with his coauthors, has developed a comprehensive mathematical theory of symmetry that allow