Interest in polyhedra runs through the whole gamut of intellectual activity from the two-year-old child who plays with wooden cubes to the mature mathematician who studies the subtleties of Branko Grünbaum's Convex polytopes (Wiley, New York, 1967). Some of the regular and semi-regular solids occur in nature as crystals, others as viruses (revealed by the electron microscope). Bees made hexagonal honey-combs long before man existed, and in human history the making of flat-faced solids (such as pyramids) is as ancient as any other kind of sculpture. The five regular solids were studied by Theætetus, Plato, Euclid, Hypsicles, and Pappus.

A considerable portion of the present book is devoted to ‘uniform’ polyhedra, which have the same arrangement of regular polygons at every corner. (Such a polyhedron is ‘regular’ if the polygons are all alike.) In any convex solid, a theorem of Euclid tells us that the angles at a corner must add up to less than 360°. After making a few models for himself, the reader will soon discover that the amount by which the angle-sum falls short of 360° is quite considerable when there are few corners (e.g. 90° for the cube, which has eight corners) but much smaller when there are many (e.g. 12° for the snub dodecahedron, which has sixty corners). This observation was fashioned into a theorem by René Descartes (1596-1650), who proved that the angular defect, added up for all the corners, always makes a total of 720°.

At about the same time, Johann Kepler (1571–1630) wrote an essay on The six-cornered snow-flake (English edition, Oxford, 1966), in which he revealed his fondness for these figures by remarking (p. 37): ‘Now among the regular solids, the first, the firstborn and father of all the rest, is the cube, and his wife, so to speak, is the octahedron, which has as many corners as the cube has faces.’