Enhanced descriptions from Syndetics:

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 allows the description and classification of symmetries in numerous geometric environments.

This richly and compellingly illustrated book addresses the phenomenological, analytical, and mathematical aspects of symmetry on three levels that build on one another and will speak to interested lay people, artists, working mathematicians, and researchers.

Reviews provided by Syndetics

CHOICE Review

This work by Conway (Princeton) and colleagues can be considered a research monograph on geometrical topology written so that anyone can read it, or a platform for pedagogically reforming group theory and so abstract algebra. Group theory studies pure symmetry--when group theory examines the symmetries of a thing, it endows the systems of symmetries (the group) with "objecthood" and lets the thing itself fade away. Students who meet groups through examples of symmetrical things often fixate on extraneous features, but a purely formal, thing-free approach to group theory makes the subject dry. This rich study of symmetrical things (that do not fade away) prepares the mind for abstract group theory. It gets somewhere, it justifies the time invested with striking results, and it develops (without using much group theory) phenomena that demand abstraction to yield their fuller meaning. The symmetrical things considered here take the form of, first, repetitive decorations of the plane, and then similarly of infinite strips, spheres, the non-Euclidean hyperbolic plane, and then higher dimensional analogs of these. Conway's insights famously simplified the foundations of these ideas in Groups, Combinatorics and Geometry, edited by M.W. Liebeck and J. Saxl (1992), but the present work offers the fullest available exposition with many new results. Summing Up: Highly recommended. All collections. D. V. Feldman University of New Hampshire