The properties of polytopes, the four-dimensional analog of polyhedra, exercise an intellectual fascination that appeals strongly to the mathematically inclined, whether they are professionals, students or amateurs. In this classic book, Professor Coxeter explores these properties in easy stages introducing the reader to complex polytopes (a beautiful generalization of regular solids derived from complex numbers) and the unexpected relationships that complex polytopes have with concepts from various branches of mathematics. In the first half of the book the author discusses magic squares, frieze patterns, kaleidoscopes, Cayley diagrams, Clifford surfaces, non-crystallographic groups, kinematics, spherical trigonometry, and algebraic geometry. Later these ideas are assembled to describe a natural generalization of the Five Platonic Solids. The fully updated second edition contains a new chapter on "Almost Regular Polytopes" and beautiful abstract art drawings. In addition, new exercises and discussions, including an introduction to Hopf fibration and real representations for two complex polyhedra, supplement the text.