Kappraff is among those who delight in discovering geometry in every corner of the world: in art, architecture, biology, music, and engineering design. His book explores the “grammar of space” through a study of similarity and symmetry, graphs, tiling, and polyhedra. Although ostensibly a treatise on “design science,” the book is primarily a catalog of fascinating geometric topics, presented with the glee of a child describing a toy collection.

Among the toys described is Fuller’s jitterbug, a flexible model of a cuboctahedron that can be deformed magically to an icosahedron, an octahedron, and a tetrahedron in sequence. Fractals, Penrose tilings, Steiner trees, origami, tensegrity, and the Poincaré plane are each touched upon. The material ranges from elementary (the golden ratio) to difficult (the Slizard and Császár polyhedra) to recent research (spiderwebs and reciprocal diagrams). Along the way are fascinating curios: how to make an icosahedron from three interpenetrating and mutually orthogonal golden rectangle index cards, how a Hamiltonian path of edges of an n-cube determines the disk moves for an n-peg Towers of Hanoi puzzle, why a steer’s horns grow in a logarithmic spiral, and why K₁₁ should be realizable as a triangulated polyhedron of genus 6 (no construction has yet been found).

Kappraff is a mathematician, and he presents elegant proofs in smaller type, presumably to permit his architecture students to skip them without guilt. Although I wonder how comprehensible students will find a text that moves from group axioms to quotient groups in just three pages, I have no doubt that they will be dazzled by this engaging compendium of geometric delights.