The third section of the book provides a classification of the three-dimensional space groups and examples of honeycombs such as the Weaire–Phelan structure. It also considers the symmetries of less familiar geometries: higher dimensional spaces, non-Euclidean spaces, and three-dimensional flat manifolds. Hyperbolic groups are used to provide a new explanation of the problem of hearing the shape of a drum. It includes the first published classification of four-dimensional convex uniform polytopes announced by Conway and Richard K. Guy in 1965, and a discussion of William Thurston 's geometrization conjecture, proved by Grigori Perelman shortly before the publication of the book, according to which all three-dimensional manifolds can be realized as symmetric spaces. One omission lamented by Jaron Lanier is the set of regular projective polytopes such as the 11-cell.