The Fifty-Nine Icosahedra

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]

(i) The faces must lie in twenty planes, viz., the bounding planes of the regular icosahedron.

(ii) All parts composing the faces must be the same in each plane, although they may be quite disconnected.

(iii) The parts included in any one plane must have trigonal symmetry, without or with reflection. This secures icosahedral symmetry for the whole solid.

(iv) The parts included in any plane must all be "accessible" in the completed solid (i.e. they must be on the "outside". In certain cases we should require models of enormous size in order to see all the outside. With a model of ordinary size, some parts of the "outside" could only be explored by a crawling insect).

(v) We exclude from consideration cases where the parts can be divided into two sets, each giving a solid with as much symmetry as the whole figure. But we allow the combination of an enantiomorphous pair having no common part (which actually occurs in just one case).

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:

λ may be 3 or 4

μ may be 7 or 8

ν may be 11 or 12

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]

Index numbers are the Crennells' unless otherwise stated:

Crennell

• In the index numbering added to the Third Edition by the Crennells, the first 32 forms (indices 1-32) are reflective models, and the last 27 (indices 33–59) are chiral with only the right-handed forms listed. This follows the order in which the stellations are depicted in the book.

Cells

• In Du Val's notation, each shell is identified in bold type, working outwards, as a, b, c, ..., h with a being the original icosahedron. Some shells subdivide into two types of cell, for example e comprises e₁ and e₂. The set f₁ further subdivides into right- and left-handed forms, respectively f₁ (plain type) and f₁ (italic). Where a stellation has all cells present within an outer shell, the outer shell is capitalised and the inner omitted, for example a + b + c + e₁ is written as Ce₁.

Faces

• All of the stellations can be specified by a stellation diagram. In the diagram shown here, the numbered colors indicate the regions of the stellation diagram which must occur together as a set, if full icosahedral symmetry is to be maintained. The diagram has 13 such sets. Some of these subdivide into chiral pairs (not shown), allowing stellations with rotational but not reflexive symmetry. In the table, faces which are seen from underneath are indicated by an apostrophe, for example 3'.

Wenninger

• The index numbers and the numbered names were allocated arbitrarily by Wenninger's publisher according to their occurrence in his book Polyhedron models and bear no relation to any mathematical sequence. Only a few of his models were of icosahedra. His names are given in shortened form, with "... of the icosahedron" left off.

Wheeler

• Wheeler found his figures, or "forms" of the icosahedron, by selecting line segments from the stellation diagram. He carefully distinguished this from Kepler's classical stellation process. Coxeter et al. ignored this distinction and referred to all of them as stellations.

Brückner

• Max Brückner made and photographed models of many polyhedra, only a few of which were icosahedra. Taf. is an abbreviation of Tafel, German for plate.

Remarks

• No. 8 is sometimes called the echidnahedron after an imagined similarity to the spiny anteater or echidna. This usage is independent of Kepler's description of his regular star polyhedra as his echidnae.

Some images illustrate the mirror-image icosahedron with the f₁ rather than the f₁ cell.

Crennell	Cells	Faces	Wenninger	Wheeler	Brückner	Remarks	Face diagram	3D
1	A	0	04 Icosahedron	1		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	99	4	Taf. IX, Fig.17
5	E	5 6 7	99	99
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	e1	3' 5	37 Twelfth stellation	99		Twelfth stellation of icosahedron
10	f1	5' 6' 9 10	99	99
11	g1	10' 12	29 Fourth stellation	21		Fourth stellation of icosahedron
12	e1f1	3' 6' 9 10	99	99
13	e1f1g1	3' 6' 9 12	99	20
14	f1g1	5' 6' 9 12	99	99
15	e2	4' 6 7	99	99
16	f2	7' 8	99	22
17	g2	8' 9'11	99	99
18	e2f2	4' 6 8	99	99
19	e2f2g2	4' 6 9' 11	99	99
20	f2g2	7' 9' 11	30 Fifth stellation	99		Fifth stellation of icosahedron
21	De1	4 5	32 Seventh stellation	10		Seventh stellation of icosahedron
22	Ef1	7 9 10	25 Compound of ten tetrahedra	8	Taf. IX, Fig. 3	Regular compound of ten tetrahedra
23	Fg1	8 9 12	31 Sixth stellation	17	Taf. X, Fig. 3	Sixth stellation of icosahedron
24	De1f1	4 6' 9 10	99	99
25	De1f1g1	4 6' 9 12	99	99
26	Ef1g1	7 9 12	28 Third stellation	9	Taf. VIII, Fig. 26	Excavated dodecahedron
27	De2	3 6 7	99	5
28	Ef2	5 6 8	99	18	Taf.IX, Fig. 20
29	Fg2	10 11	33 Eighth stellation	14		Eighth stellation of icosahedron
30	De2f2	3 6 8	34 Ninth stellation	13		Medial triambic icosahedron or Great triambic icosahedron
31	De2f2g2	3 6 9' 11	99	99
32	Ef2g2	5 6 9' 11	99	99
33	f1	5' 6' 9 10	35 Tenth stellation	99		Tenth stellation of icosahedron
34	e1f1	3' 5 6' 9 10	36 Eleventh stellation	99		Eleventh stellation of icosahedron
35	De1f1	4 5 6' 9 10	99	99
36	f1g1	5' 6' 9 10' 12	99	99
37	e1f1g1	3' 5 6' 9 10' 12	39 Fourteenth stellation	99		Fourteenth stellation of icosahedron
38	De1f1g1	4 5 6' 9 10' 12	99	99
39	f1g2	5' 6' 8' 9' 10 11	99	99
40	e1f1g2	3' 5 6' 8' 9' 10 11	99	99
41	De1f1g2	4 5 6' 8' 9' 10 11	99	99
42	f1f2g2	5' 6' 7' 9' 10 11	99	99
43	e1f1f2g2	3' 5 6' 7' 9' 10 11	99	99
44	De1f1f2g2	4 5 6' 7' 9' 10 11	99	99
45	e2f1	4' 5' 6 7 9 10	40 Fifteenth stellation	99		Fifteenth stellation of icosahedron
46	De2f1	3 5' 6 7 9 10	99	99
47	Ef1	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	e2f1g1	4' 5' 6 7 9 10' 12	99	99
49	De2f1g1	3 5' 6 7 9 10' 12	99	99
50	Ef1g1	5 6 7 9 10' 12	99	99
51	e2f1f2	4' 5' 6 8 9 10	38 Thirteenth stellation	99		Thirteenth stellation of icosahedron
52	De2f1f2	3 5' 6 8 9 10	99	99
53	Ef1f2	5 6 8 9 10	99	15 (16: left handed)
54	e2f1f2g1	4' 5' 6 8 9 10' 12	99	99
55	De2f1f2g1	3 5' 6 8 9 10' 12	99	99
56	Ef1f2g1	5 6 8 9 10' 12	99	99
57	e2f1f2g2	4' 5' 6 9' 10 11	99	99
58	De2f1f2g2	3 5' 6 9' 10 11	99	99
59	Ef1f2g2	5 6 9' 10 11	99	99