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.
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":^{[1]}
Rules (i) to (iii) are symmetry requirements for the face planes. Rule (iv) excludes buried holes, to ensure that no two stellations look outwardly identical. Rule (v) prevents any disconnected compound of simpler stellations.
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 (see below). The Greek symbols represent sets of possible alternatives:
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'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.^{[2]}
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.
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 (which at that time were all thought to be Flather's). Corrections to this edition have been published online.^{[3]}
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.
Index numbers are the Crennells' unless otherwise stated:
Crennell
Cells
Faces
Wenninger
Wheeler
Brückner
Remarks
Some images illustrate the mirror-image icosahedron with the f_{1} rather than the f_{1} cell.
Crennell | Cells | Faces | Wenninger | Wheeler | Brückner | Remarks | Face diagram | 3D |
---|---|---|---|---|---|---|---|---|
1 | A | 0 | Icosahedron |
41 | The Platonic icosahedron | |||
2 | B | 1 | 26 Triakis icosahedron |
2 | Taf. VIII, Fig. 2 | First stellation of the icosahedron, small triambic icosahedron, or Triakisicosahedron |
||
3 | C | 2 | 23 Compound of five octahedra |
3 | Taf. IX, Fig. 6 | Regular compound of five octahedra | ||
4 | D | 3 4 | 4 | Taf. IX, Fig.17 | ||||
5 | E | 5 6 7 | ||||||
6 | F | 8 9 10 | 27 Second stellation |
19 | Second stellation of icosahedron | |||
7 | G | 11 12 | 41 Great icosahedron |
11 | Taf. XI, Fig. 24 | Great icosahedron | ||
8 | H | 13 | 42 Final stellation |
12 | Taf. XI, Fig. 14 | Final stellation of the icosahedron or Echidnahedron | ||
9 | e_{1} | 3' 5 | 37 Twelfth stellation |
Twelfth stellation of icosahedron | ||||
10 | f_{1} | 5' 6' 9 10 | ||||||
11 | g_{1} | 10' 12 | 29 Fourth stellation |
21 | Fourth stellation of icosahedron | |||
12 | e_{1}f_{1} | 3' 6' 9 10 | ||||||
13 | e_{1}f_{1}g_{1} | 3' 6' 9 12 | 20 | |||||
14 | f_{1}g_{1} | 5' 6' 9 12 | ||||||
15 | e_{2} | 4' 6 7 | ||||||
16 | f_{2} | 7' 8 | 22 | |||||
17 | g_{2} | 8' 9'11 | ||||||
18 | e_{2}f_{2} | 4' 6 8 | ||||||
19 | e_{2}f_{2}g_{2} | 4' 6 9' 11 | ||||||
20 | f_{2}g_{2} | 7' 9' 11 | 30 Fifth stellation |
Fifth stellation of icosahedron | ||||
21 | De_{1} | 4 5 | 32 Seventh stellation |
10 | Seventh stellation of icosahedron | |||
22 | Ef_{1} | 7 9 10 | 25 Compound of ten tetrahedra |
8 | Taf. IX, Fig. 3 | Regular compound of ten tetrahedra | ||
23 | Fg_{1} | 8 9 12 | 31 Sixth stellation |
17 | Taf. X, Fig. 3 | Sixth stellation of icosahedron | ||
24 | De_{1}f_{1} | 4 6' 9 10 | ||||||
25 | De_{1}f_{1}g_{1} | 4 6' 9 12 | ||||||
26 | Ef_{1}g_{1} | 7 9 12 | 28 Third stellation |
9 | Taf. VIII, Fig. 26 | Excavated dodecahedron | ||
27 | De_{2} | 3 6 7 | 5 | |||||
28 | Ef_{2} | 5 6 8 | 18 | Taf.IX, Fig. 20 | ||||
29 | Fg_{2} | 10 11 | 33 Eighth stellation |
14 | Eighth stellation of icosahedron | |||
30 | De_{2}f_{2} | 3 6 8 | 34 Ninth stellation |
13 | Medial triambic icosahedron or Great triambic icosahedron |
|||
31 | De_{2}f_{2}g_{2} | 3 6 9' 11 | ||||||
32 | Ef_{2}g_{2} | 5 6 9' 11 | ||||||
33 | f_{1} | 5' 6 9 10 | 35 Tenth stellation |
Tenth stellation of icosahedron | ||||
34 | e_{1}f_{1} | 3 5 6 9 10 | 36 Eleventh stellation |
Eleventh stellation of icosahedron | ||||
35 | De_{1}f_{1} | 4 5 6 9 10 | ||||||
36 | f_{1}g_{1} | 5' 6 9 10 12 | ||||||
37 | e_{1}f_{1}g_{1} | 3 5 6 9 10 12 | 39 Fourteenth stellation |
Fourteenth stellation of icosahedron | ||||
38 | De_{1}f_{1}g_{1} | 4 5 6 9 10 12 | ||||||
39 | f_{1}g_{2} | 5' 6 8 9 10 11 | ||||||
40 | e_{1}f_{1}g_{2} | 3 5 6 8 9 10 11 | ||||||
41 | De_{1}f_{1}g_{2} | 4 5 6 8 9 10 11 | ||||||
42 | f_{1}f_{2}g_{2} | 5' 6 7 9 10 11 | ||||||
43 | e_{1}f_{1}f_{2}g_{2} | 3 5 6 7 9 10 11 | ||||||
44 | De_{1}f_{1}f_{2}g_{2} | 4 5 6 7 9 10 11 | ||||||
45 | e_{2}f_{1} | 4 5 6 7 9 10 | 40 Fifteenth stellation |
Fifteenth stellation of icosahedron | ||||
46 | De_{2}f_{1} | 3 5 6 7 9 10 | ||||||
47 | Ef_{1} | 5 6 7 9 10 | 24 Compound of five tetrahedra |
7 (6: left handed) |
Taf. IX, Fig. 11 | Regular Compound of five tetrahedra (right handed) | ||
48 | e_{2}f_{1}g_{1} | 4 5 6 7 9 10 12 | ||||||
49 | De_{2}f_{1}g_{1} | 3 5 6 7 9 10 12 | ||||||
50 | Ef_{1}g_{1} | 5 6 7 9 10 12 | ||||||
51 | e_{2}f_{1}f_{2} | 4 5 6 8 9 10 | 38 Thirteenth stellation |
Thirteenth stellation of icosahedron | ||||
52 | De_{2}f_{1}f_{2} | 3 5 6 8 9 10 | ||||||
53 | Ef_{1}f_{2} | 5 6 8 9 10 | 15 (16: left handed) |
|||||
54 | e_{2}f_{1}f_{2}g_{1} | 4 5 6 8 9 10 12 | ||||||
55 | De_{2}f_{1}f_{2}g_{1} | 3 5 6 8 9 10 12 | ||||||
56 | Ef_{1}f_{2}g_{1} | 5 6 8 9 10 12 | ||||||
57 | e_{2}f_{1}f_{2}g_{2} | 4 5 6 9 10 11 | ||||||
58 | De_{2}f_{1}f_{2}g_{2} | 3 5 6 9 10 11 | ||||||
59 | Ef_{1}f_{2}g_{2} | 5 6 9 10 11 |