Winner, Euler Book Prize, awarded by the Mathematical Association of America.With over 200 full color photographs, this non-traditional, tactile introduction to non-Euclidean geometries also covers early development of geometry and connections between geometry, art, nature, and sciences.For the crafter or would-be crafter, there are detailed instructions for how to crochet various geometric models and how to use them in explorations.New to the 2nd Edition; Daina Taimina discusses her own adventures with the hyperbolic planes as well as the experiences of some of her readers.Includes recent applications of hyperbolic geometry such as medicine, architecture, fashion & quantum computing.