The Fifty-Nine Icosahedra


The Fifty-Nine Icosahedra is a book written and illustrated by H. S. M. Coxeter, P. Du Val, H. T. Flather and J. F. Petrie. It enumerates certain stellations of the regular convex or Platonic icosahedron, according to a set of rules put forward by J. C. P. Miller.
First published by the University of Toronto in 1938, a Second Edition reprint by Springer-Verlag followed in 1982. Tarquin's 1999 Third Edition included new reference material and photographs by K. and D. Crennell.

Authors' contributions

Miller's rules

Although Miller did not contribute to the book directly, he was a close colleague of Coxeter and Petrie. His contribution is immortalised in his set of rules for defining which stellation forms should be considered "properly significant and distinct":
Rules to are symmetry requirements for the face planes. Rule excludes buried holes, to ensure that no two stellations look outwardly identical. Rule prevents any disconnected compound of simpler stellations.

Coxeter

Coxeter was the main driving force behind the work. He carried out the original analysis based on Miller's rules, adopting a number of techniques such as combinatorics and abstract graph theory whose use in a geometrical context was then novel.
He observed that the stellation diagram comprised many line segments. He then developed procedures for manipulating combinations of the adjacent plane regions, to formally enumerate the combinations allowed under Miller's rules.
His graph, reproduced here, shows the connectivity of the various faces identified in the stellation diagram. The Greek symbols represent sets of possible alternatives:

Du Val

Du Val devised a symbolic notation for identifying sets of congruent cells, based on the observation that they lie in "shells" around the original icosahedron. Based on this he tested all possible combinations against Miller's rules, confirming the result of Coxeter's more analytical approach.

Flather

Flather's contribution was indirect: he made card models of all 59. When he first met Coxeter he had already made many stellations, including some "non-Miller" examples. He went on to complete the series of fifty-nine, which are preserved in the mathematics library of Cambridge University, England. The library also holds some non-Miller models, but it is not known whether these were made by Flather or by Miller's later students.

Petrie

John Flinders Petrie was a lifelong friend of Coxeter and had a remarkable ability to visualise four-dimensional geometry. He and Coxeter had worked together on many mathematical problems. His direct contribution to the fifty nine icosahedra was the exquisite set of three-dimensional drawings which provide much of the fascination of the published work.

The Crennells

For the Third Edition, Kate and David Crennell reset the text and redrew the diagrams. They also added a reference section containing tables, diagrams, and photographs of some of the Cambridge models. Corrections to this edition have been published online.

List of the fifty nine icosahedra

Before Coxeter, only Brückner and Wheeler had recorded any significant sets of stellations, although a few such as the great icosahedron had been known for longer. Since publication of The 59, Wenninger published instructions on making models of some; the numbering scheme used in his book has become widely referenced, although he only recorded a few stellations.

Table of the fifty-nine icosahedra

Some images illustrate the mirror-image icosahedron with the f1 rather than the f1 cell.
CrennellCellsFacesWenningerWheelerBrücknerRemarksFace diagram3D
1A004
Icosahedron		1The Platonic icosahedron
2B126
Triakis icosahedron		2First stellation of the icosahedron,
small triambic icosahedron,
or Triakisicosahedron
3C223
Compound of five octahedra		3Regular compound of five octahedra
4D3 44
5E5 6 7
6F8 9 1027
Second stellation		19Second stellation of icosahedron
7G11 1241
Great icosahedron		11Great icosahedron
8H1342
Final stellation		12Final stellation of the icosahedron or Echidnahedron
9e13' 537
Twelfth stellation		Twelfth stellation of icosahedron
10f15' 6' 9 10
11g110' 1229
Fourth stellation		21Fourth stellation of icosahedron
12e1f13' 6' 9 10
13e1f1g13' 6' 9 1220
14f1g15' 6' 9 12
15e24' 6 7
16f27' 822
17g28' 9'11
18e2f24' 6 8
19e2f2g24' 6 9' 11
20f2g27' 9' 1130
Fifth stellation		Fifth stellation of icosahedron
21De14 532
Seventh stellation		10Seventh stellation of icosahedron
22Ef17 9 1025
Compound of ten tetrahedra		8Regular compound of ten tetrahedra
23Fg18 9 1231
Sixth stellation		17Sixth stellation of icosahedron
24De1f14 6' 9 10
25De1f1g14 6' 9 12
26Ef1g17 9 1228
Third stellation		9Excavated dodecahedron
27De23 6 75
28Ef25 6 818
29Fg210 1133
Eighth stellation		14Eighth stellation of icosahedron
30De2f23 6 834
Ninth stellation		13Medial triambic icosahedron or
Great triambic icosahedron
31De2f2g23 6 9' 11
32Ef2g25 6 9' 11
33f15' 6 9 1035
Tenth stellation		Tenth stellation of icosahedron
34e1f13 5 6 9 1036
Eleventh stellation		Eleventh stellation of icosahedron
35De1f14 5 6 9 10
36f1g15' 6 9 10 12
37e1f1g13 5 6 9 10 1239
Fourteenth stellation		Fourteenth stellation of icosahedron
38De1f1g14 5 6 9 10 12
39f1g25' 6 8 9 10 11
40e1f1g23 5 6 8 9 10 11
41De1f1g24 5 6 8 9 10 11
42f1f2g25' 6 7 9 10 11
43e1f1f2g23 5 6 7 9 10 11
44De1f1f2g24 5 6 7 9 10 11
45e2f14 5 6 7 9 1040
Fifteenth stellation		Fifteenth stellation of icosahedron
46De2f13 5 6 7 9 10
47Ef15 6 7 9 1024
Compound of five tetrahedra		7
Regular Compound of five tetrahedra
48e2f1g14 5 6 7 9 10 12
49De2f1g13 5 6 7 9 10 12
50Ef1g15 6 7 9 10 12
51e2f1f24 5 6 8 9 1038
Thirteenth stellation		Thirteenth stellation of icosahedron
52De2f1f23 5 6 8 9 10
53Ef1f25 6 8 9 1015
54e2f1f2g14 5 6 8 9 10 12
55De2f1f2g13 5 6 8 9 10 12
56Ef1f2g15 6 8 9 10 12
57e2f1f2g24 5 6 9 10 11
58De2f1f2g23 5 6 9 10 11
59Ef1f2g25 6 9 10 11