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.
It was first published by the University of Toronto in 1938, and 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.

Stellating polyhedra

In this book a polyhedron is stellated by extending the face planes of a polyhedron until they meet again to form a new polyhedron or compound. When the face planes of the polyhedron are extended indefinitely the space around the polyhedron is divided into unbounded sub spaces and often a number of bounded polyhedrons or cells. Different sets of cells yield different stellations.
For a symmetrical polyhedron, these cells will fall into groups of congruent cells, or sets – we say that the cells in such a set are of the same type.
This can still lead to a large number of possible forms, so further criteria are imposed to reduce the set to those stellations that are significant and distinct in some way.
Among the Platonic solids, the tetrahedron and cube have no stellations, the octahedron has one, the dodecahedron has three and the icosahedron has a much larger number.

Earlier publications on stellated icosahedra

The Fifty-Nine Icosahedra is not the first work about stellated icosahedra.
In 1809 Louis Poinsot discovered the first recognised examples, the great icosahedron and the great dodecahedron, completing the set of what are nowadays known as the regular star or Kepler–Poinsot polyhedra.
In 1876 Edmund Hess used stellation diagrams and discovered the remaining mainline stellated icosahedra.
In 1900 Max Brückner described and photographed many stellated icosahedra in his book Vielecke und Vielflache: Theorie und Geschichte.
In 1924 A. Harry Wheeler gave a talk as an Invited Speaker of the ICM in 1924 at Toronto.
In his talk he presented the method of selecting regions of the stellation diagram and combining their cells to form new polyhedral figures. Wheeler included hollow polyhedra and sets of discrete cells.
Wheeler was initially to be a co-author of The Fifty-Nine Icosahedra, but he objected to Coxeter's approach, which he found so “involved and clumsy that I did not want to have anything to do with it.... Coxeter has a way of taking a subject and tying it up into knots in such a way that I find it quite difficult to follow him and some times to even make sense.”

Authors' contributions

Miller's rules

Although J. C. P. 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 stellations 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 all the extended face planes together cut the space around the icosahedron in many different finite three dimensional regions that he called cells.
A segment between a point in a cell and the centre of the icosahedron intersects a number of face planes; this number is the power of the point and the cell.
All cells with the same power form a shell. The inner icosahedron is named A, the shell with power 1 b, the shell with power 2 c, and so on.
If all cells of a shell are congruent, they are named as the shell itself; if there are different non-congruent cells in a shell, they are numbered like e1 and e2. If an enantiomorphic pair of cells is in a shell, if both are referred then f1 is used, if only one of them is used one is in roman and the other in italic like f1 and f1.
For example, there are 3 kinds of cells in the layer with power 5 : f1, f1 and f2. f1 is used if only f1, and f1 are used.
A stellation consisting of a complete shell and all cells interior to it is named after the outer shell, capitalised, for example B for A + b and De1 for A + b + c + d+ e1.
Any combination of cells form a stellated icosahedron, except that the last two of Millers conditions rule out many combinations.
With this scheme, Du Val tested all possible combinations against Miller's rules, confirming the result of Coxeter's 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 numbered the icosahedra, 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

Several more restrictive categories of stellated polyhedra have been identified, some are easily recognised: mainline stellations are the stellations whose name consist of a single capital letterA till Hfully supported stellations: stellations where there are no overhangs, and all visible parts of a face can be seen from the same side. these are those stellations where in the faces column there is no face that has an apostrophe, for example 3.
The only Kepler–Poinsot polyhedron in the list is the great icosahedron G.
The Great stellated dodecahedron is an edge-stellated icosahedron but only face-stellated icosahedra are in the list and it is therefore not included in the list.
Some images illustrate the mirror-image icosahedron with the f₁ rather than the f₁ cell.

Index	Cells	Faces	Wenninger	Wheeler	Brückner	Remarks	Face diagram	3D
1	A	0	04 Icosahedron	1 Regular convex triangular		Regular icosahedron, a Platonic solid
2	B	1	26 Triakis icosahedron	2 Hexagonal		First stellation of the icosahedron, small triambic icosahedron, or triakisicosahedron
3	C	2	23 Compound of five octahedra	3 Five intersecting octahedra		Regular compound of five octahedra
4	D	3 4		4 Nonagonal
5	E	5 6 7
6	F	8 9 10	27 Second stellation	19 triple 13+6+7		Second stellation of icosahedron
7	G	11 12	41 Great icosahedron	11 Regular star Poinsot		Great icosahedron
8	H	13	42 Final stellation	12 Complete		Final stellation of the icosahedron or echidnahedron
9	e₁	3' 5	37 Twelfth stellation			Twelfth stellation of icosahedron
10	f₁	5' 6' 9 10
11	g₁	10' 12	29 Fourth stellation	21 Discrete skeleton		Fourth stellation of icosahedron
12	e₁f₁	3' 6' 9 10
13	e₁f₁g₁	3' 6' 9 12		20 Hollow, labyrinth
14	f₁g₁	5' 6' 9 12
15	e₂	4' 6 7
16	f₂	7' 8		22 Discrete twelve-pointed, crown-rimmed group
17	g₂	8' 9'11
18	e₂f₂	4' 6 8
19	e₂f₂g₂	4' 6 9' 11
20	f₂g₂	7' 9' 11	30 Fifth stellation			Fifth stellation of icosahedron
21	De₁	4 5	32 Seventh stellation	10 Twenty-pointed, six-edged		Seventh stellation of icosahedron
22	Ef₁	7 9 10	25 Compound of ten tetrahedra	8 Ten intersecting tetrahedra		Regular compound of ten tetrahedra
23	Fg₁	8 9 12	31Sixth stellation	17 Double 13+ 9		Sixth stellation of icosahedron
24	De₁f₁	4 6' 9 10
25	De₁f₁g₁	4 6' 9 12
26	Ef₁g₁	7 9 12	28Third stellation	9 Mobius		Excavated dodecahedron
27	De₂	3 6 7		5 Archimedian variety of no 13
28	Ef₂	5 6 8		18 Double 13 + 10
29	Fg₂	10 11	33 Eighth stellation	14 Archimedian variety 11		Eighth stellation of icosahedron
30	De₂f₂	3 6 8	34 Ninth stellation	13 Kite archimedian variety no 11		Medial triambic icosahedron or Great triambic icosahedron
31	De₂f₂g₂	3 6 9' 11
32	Ef₂g₂	5 6 9' 11
33	f₁	5' 6 9 10	35 Tenth stellation			Tenth stellation of icosahedron
34	e₁f₁	3 5 6 9 10	36 Eleventh stellation			Eleventh stellation of icosahedron
35	De₁f₁	4 5 6 9 10
36	f₁g₁	5' 6 9 10 12
37	e₁f₁g₁	3 5 6 9 10 12	39 Fourteenth stellation			Fourteenth stellation of icosahedron
38	De₁f₁g₁	4 5 6 9 10 12
39	f₁g₂	5' 6 8 9 10 11
40	e₁f₁g₂	3 5 6 8 9 10 11
41	De₁f₁g₂	4 5 6 8 9 10 11
42	f₁f₂g₂	5' 6 7 9 10 11
43	e₁f₁f₂g₂	3 5 6 7 9 10 11
44	De₁f₁f₂g₂	4 5 6 7 9 10 11
45	e₂f₁	4 5 6 7 9 10	40 Fifteenth stellation			Fifteenth stellation of icosahedron
46	De₂f₁	3 5 6 7 9 10
47	Ef₁	5 6 7 9 10	24 Compound of five tetrahedra	7: right handed 6: left handed Five intersecting Tetrahedra		Regular Compound of five tetrahedra
48	e₂f₁g₁	4 5 6 7 9 10 12
49	De₂f₁g₁	3 5 6 7 9 10 12
50	Ef₁g₁	5 6 7 9 10 12
51	e₂f₁f₂	4 5 6 8 9 10	38 Thirteenth stellation			Thirteenth stellation of icosahedron
52	De₂f₁f₂	3 5 6 8 9 10
53	Ef₁f₂	5 6 8 9 10		15: right handed 16: left handed Double 13+7
54	e₂f₁f₂g₁	4 5 6 8 9 10 12
55	De₂f₁f₂g₁	3 5 6 8 9 10 12
56	Ef₁f₂g₁	5 6 8 9 10 12
57	e₂f₁f₂g₂	4 5 6 9 10 11
58	De₂f₁f₂g₂	3 5 6 9 10 11
59	Ef₁f₂g₂	5 6 9 10 11

Subsequent debate

There has subsequently been some debate on Miller's rules, with some writers questioning them. Flather himself made and exhibited models which were "non-Miller". Bridge obtained stellations of the icosahedron by dualising facettings of the dodecahedron, noting the significance of internal structure in distinguishing between stellations which Miller's rules treat as identical. Hudson and Kingston adopted a reduced rule set. Inchbald noted two non-Miller stellations, and went on to discuss various related issues.