*Mathematical Models* (Cundy and Rollett)

*https://en.wikipedia.org/wiki/Mathematical_Models_(Cundy_and_Rollett)*

* Mathematical Models* is a book on the construction of physical models of mathematical objects for educational purposes. It was written by
Martyn Cundy and A. P. Rollett, and published by the
Clarendon Press in 1951,

^{ [1]}

^{ [2]}

^{ [3]}

^{ [4]}

^{ [5]}

^{ [6]}with a second edition in 1961.

^{ [2]}

^{ [7]}Tarquin Publications published a third edition in 1981.

^{ [8]}

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.^{
[9]}

## 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.^{
[5]} The media used for the constructions described in the book include "paper, cardboard, plywood, plastics, wire, string, and sheet metal".^{
[1]}

The second chapter concerns plane geometry, and includes material on the
golden ratio,^{
[5]} the
Pythagorean theorem,^{
[6]}
dissection problems, the
mathematics of paper folding,
tessellations, and
plane curves, which are constructed by stitching, by graphical methods, and by mechanical devices.^{
[1]}

The third chapter, and the largest part of the book, concerns
polyhedron models,^{
[1]} made from cardboard or plexiglass.^{
[6]} It includes information about the
Platonic solids,
Archimedean solids, their
stellations and
duals,
uniform polyhedron compounds, and
deltahedra.^{
[1]}

The fourth chapter is on additional topics in
solid geometry^{
[5]} and
curved surfaces, particularly
quadrics^{
[1]} 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.^{
[1]}^{
[3]} Also included are
sphere packings.^{
[4]} The models in this chapter are constructed as the boundaries of solid objects, via two-dimensional paper cross-sections, and by
string figures.^{
[1]}

The fifth chapter, and the final one of the first edition, includes mechanical apparatus including
harmonographs and
mechanical linkages,^{
[1]} the
bean machine and its demonstration of the
central limit theorem, and analogue computation using
hydrostatics.^{
[3]} The second edition expands this chapter, and adds another chapter on computational devices such as the
differential analyser of
Vannevar Bush.^{
[7]}

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.^{
[1]}

## Audience and reception

At the time they wrote the book, Cundy and Rollett were
sixth form teachers in the UK,^{
[1]}^{
[4]} and they intended the book to be used by mathematics students and teachers for educational activities at that level.^{
[1]}^{
[6]} However, it may also be enjoyed by a general audience of mathematics enthusiasts.^{
[3]}

Reviewer Michael Goldberg notes some minor errors in the book's historical credits and its notation, and writes that for American audiences some of the British terminology may be unfamiliar, but concludes that it could still be valuable for students and teachers. Stanley Ogilvy complains about the inconsistent level of rigor of the mathematical descriptions, with some proofs given and others omitted, for no clear reason, but calls this issue minor and in general calls the book's presentation excellent.
Dirk ter Haar is more enthusiastic, recommending it to anyone interested in mathematics, and suggesting that it should be required for mathematics classrooms.^{
[3]} Similarly, B. J. F. Dorrington recommends it to all mathematical libraries,^{
[5]} and The Basic Library List Committee of the
Mathematical Association of America has given it their strong recommendation for inclusion in undergraduate mathematics libraries.^{
[8]} By the time of its second edition, H. S. M. Coxeter states that *Mathematical Models* had become "well known".^{
[7]}

## References

- ^
^{a}^{b}^{c}^{d}^{e}^{f}^{g}^{h}^{i}^{j}^{k}^{l}Goldberg, M., "Review of 1st ed.",*Mathematical Reviews*, MR 0049560 - ^
^{a}^{b}Müller, H. R., "Review of 1st ed.",*zbMATH*(in German), Zbl 0047.38807; 2nd ed., Zbl 0095.38001 - ^
^{a}^{b}^{c}^{d}^{e}ter Haar, D. (March 1953), "Briefly reviewed (review of 1st ed.)",*The Scientific Monthly*,**76**(3): 188–189, JSTOR 20668 - ^
^{a}^{b}^{c}Stone, Abraham (April 1953), "Review of 1st ed.",*Scientific American*,**188**(4): 110, JSTOR 24944205 - ^
^{a}^{b}^{c}^{d}^{e}Dorrington, B. J. F. (September 1953), "Review of 1st ed.",*The Mathematical Gazette*,**37**(321): 223, doi: 10.2307/3608314, JSTOR 3608314 - ^
^{a}^{b}^{c}^{d}Ogilvy, C. Stanley (November 1959), "Review of 1st ed.",*The Mathematics Teacher*,**52**(7): 577–578, JSTOR 27956015 - ^
^{a}^{b}^{c}Coxeter, H. S. M. (December 1962), "Review of 2nd ed.",*The Mathematical Gazette*,**46**(358): 331, doi: 10.2307/3611791, JSTOR 3611791 - ^
^{a}^{b}"Mathematical Models (3rd ed.; listing with no review)",*MAA Reviews*, Mathematical Association of America, retrieved 2020-09-09 **^**Popko, Edward S. (2012), "6.4.1 Cundy–Rollett Symbols",*Divided Spheres: Geodesics and the Orderly Subdivision of the Sphere*, Boca Raton, Florida: CRC Press, doi: 10.1201/b12253-22, ISBN 978-1-4665-0429-5, MR 2952780