The Beauty of Geometry: Twelve Essays

Written by a distinguished mathematician, the dozen absorbing essays in this versatile volume offer both supplementary classroom material and pleasurable reading for the mathematically inclined. The essays promise to encourage readers in the further study of elementary geometry, not just for its own sake, but also for its broader applications, which receive a full and engaging treatment. Beginning with an analytic approach, the author reviews the functions of Schlafli and Lobatschefsky and discusses number theory in a dissertation on integral Cayley numbers. A detailed examination of group theory includes discussion of Wythoff's construction for uniform polytopes, as well as a chapter on regular skew polyhedra in three and four dimensions and their topological analogues. A profile of self-dual configurations and regular graphs introduces elements of graph theory, followed up with a chapter on twelve points in PG (5, 3) with 95040 self-transformations. Discussion of an upper bound for the number of equal nonoverlapping spheres that can touch another same-sized sphere develops aspects of communication theory, while relativity theory is explored in a chapter on reflected light signals. Additional topics include the classification of zonohedra by means of projective diagrams, arrangements of equal spheres in non-Euclidean spaces, and regular honeycombs in hyperbolic space. Stimulating and thought-provoking, this collection is sure to interest students, mathematicians, and any math buff with its lucid treatment of geometry and the crucial role geometry can play in a wide range of mathematical applications.