The Fifty-Nine Icosahedra

The Fifty-Nine Icosahedra, third revised edition

Third Revised Edition, by H S M Coxeter, P Du Val (Patrick Du Val) and H Flather.

The book first appeared in 1938 from the University of Toronto Press, with J F Petrie as fourth author. Springer-Verlag brought out a second edition in 1982.

For this edition, the plans and illustrations of all 59 of the stellations of the icosahedron were redrawn, and Professor Coxeter wrote a new introduction. This edition has now been revised and updated. It also includes colour plates for the first time, by Kate and David Crennell.

• Authors: H S M Coxeter, P Du Val, H Flather
• Edition: Third Revised Edition, revised and updated
• Contents: redrawn plans and illustrations of all 59 stellations
• New: colour plates by Kate and David Crennell
• Introduction: Professor Coxeter. Preface to the Third Edition: Richard Guy

Using the plates to draw and build paper models

Stellation starts with the regular icosahedron: its faces are extended until they meet again, which produces new stellated solids. The starting shape has 20 triangular faces and 12 vertices, and its dual is the regular dodecahedron.

Among the results are the great icosahedron and compounds of five octahedra, five tetrahedra and ten tetrahedra. Many readers will want to use the plates and descriptions in The Fifty-Nine Icosahedra to draw some or all of these stellations, or to make paper models of them.

Hands-on, the idea clicks when you trace a plan and then fold the matching model. Doing both gives you a greater understanding of the stellation process.

Who this Coxeter classic suits best

It is best suited to sixth-form teachers, maths clubs and independent learners who already know the Platonic solids. For a thorough understanding of stellation and splendid examples of beautiful polyhedra, Tarquin sees this book as a valuable addition to any mathematics library.

A natural next step is to choose one of the 59 icosahedra from the colour plates. Draw it first, then build it as a paper model.

From the preface to the third edition

As Richard Guy put it in the Preface to the Third Edition: “We mathematicians know how beautiful our subject is, but are often frustrated when we try to explain this to the man-in-the-street. But here at least is one place where we can simply say, ‘Look!’”