Mathematical Models (Cundy and Rollett)

The vertex configuration of a uniform polyhedron, a generalization of the Schläfli symbol that describes the pattern of polygons surrounding each vertex, was devised in this book as a way to name the Archimedean solids, and has sometimes been called the Cundy–Rollett symbol as a nod to this origin.

Topics

The first edition of the book had five chapters, including its introduction which discusses model-making in general and the different media and tools with which one can construct models. The media used for the constructions described in the book include "paper, cardboard, plywood, plastics, wire, string, and sheet metal".

The second chapter concerns plane geometry, and includes material on the golden ratio, the Pythagorean theorem, dissection problems, the mathematics of paper folding, tessellations, and plane curves, which are constructed by stitching, by graphical methods, and by mechanical devices.

The third chapter, and the largest part of the book, concerns polyhedron models, made from cardboard or plexiglass. It includes information about the Platonic solids, Archimedean solids, their stellations and duals, uniform polyhedron compounds, and deltahedra.

The fourth chapter is on additional topics in solid geometry and curved surfaces, particularly quadrics but also including topological manifolds such as the torus, Möbius strip and Klein bottle, and physical models helping to visualize the map coloring problem on these surfaces. Also included are sphere packings. The models in this chapter are constructed as the boundaries of solid objects, via two-dimensional paper cross-sections, and by string figures.

The fifth chapter, and the final one of the first edition, includes mechanical apparatus including harmonographs and mechanical linkages, the bean machine and its demonstration of the central limit theorem, and analogue computation using hydrostatics. The second edition expands this chapter, and adds another chapter on computational devices such as the differential analyser of Vannevar Bush.

Much of the material on polytopes was based on the book Regular Polytopes by H. S. M. Coxeter, and some of the other material has been drawn from resources previously published in 1945 by the National Council of Teachers of Mathematics.