*The main text of this article is reproduced by kind permission
from *The Mathematical Gazette* 86, July 2002, p.p. 208-215.*

*A few minor corrections have been made:
o g_{1} comprises 30 bipyramids, not hexahedra.
o Face diagram 11 12 comprises nine parts of course, not six.
o A few minor typos.*

Just over sixty years ago Coxeter, Du Val, Flather and Petrie wrote *The
fifty-nine icosahedra*. A new edition, with redrawn illustrations and
useful additional material, has recently been published [1], making this a
convenient moment to see if there is anything new to say. Coxeter once
described the work as reflecting a certain 'youthful exuberance,' so do
not be too surprised at what we find.

Stellations of a polyhedron are obtained by extending some of its edges or faces
until they intersect at a distance from the original polyhedron. One way
of studying stellations is to consider the planes in which the faces of
the polyhedron lie, that is, its face planes. The face planes of the regular
icosahedron intersect eachother (see Appendix) to dissect space into
numerous regions, of which 473 are finite cells. These cells come in just
12 shapes which form layers around the original icosahedron, itself the
innermost cell. The set of cells of a given shape comprises part or all of
a layer, with icosahedral symmetry. The various stellations can be
obtained by selecting different combinations of these cell sets. Because
there are 12 types of cell and we are not interested in the 'empty'
combination, there are 2^{12} - 1 = 4,095
possible combinations.

To distinguish apart the combinations, and hence also the stellations, Du Val developed a symbolic notation based on the cells present: I use Du Val's notation here. I will also occasionally borrow the habit of referring to a stellation of the regular icosahedron simply as an icosahedron.

Many of the 4,095 combinations do not qualify as stellations, for
various reasons. *The fifty-nine icosahedra* (I will tend to refer
to it simply as *The 59*) sought to identify and enumerate the
stellations. To give some guidance to the investigators, their tutor JCP
Miller proposed five rules:

(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, with or without 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) together secure icosahedral symmetry for the
stellation. (iv) excludes any combination having one or more fully
enclosed cavities, so ensuring that every stellation has a unique outward
appearance. Lastly (v) forbids disconnected concentric compounds of
simpler stellations: *The 59* interpreted it as also forbidding
vertex-connected or edge-connected concentric compounds, though the rule
as stated is ambiguous. Application of these rules led to the enumeration
of 58 stellations which together with the original icosahedron make up the
59 icosahedra described.

Some of the resulting stellations are evidently unsatisfactory. In
particlular **f _{2}**, shown in Figure 1, comprises twelve
quite disconnected trapezohedra floating in space. As a loose collection
of polyhedra it would appear to go against the spirit, if not the letter,
of rule (v) which attempted to ban compounds of separate parts.

Figure 1 f _{2} |
Figure 2 g_{1} |

Wheeler [2] had earlier discovered **g _{1}** and

To answer the first half of this question we need to see how
*The 59* approached the idea of a polyhedron.

As a mathematical object a polyhedron can be approached as a solid block of space, or as a collection of faces, edges, or points (vertices). The same collection of edges or of vertices can be shared by several polyhedra of differing external appearance, so these approaches are not helpful here. The choice between faces or solid space is less obvious but there is an important difference in that a solid block of space has no internal structure. The importance of internal structure has been pointed out by Cromwell [3] among others.

The great icosahedron is stellation **G**. The associated face
diagram **11** **12** in *The 59* (Figure 3)
comprises six irregular-shaped, discontinuous parts remaining wholly
external to the solid. From it we can construct the vertex figure (Figure 4)
(The vertex figure can be loosely thought of as the polygon exposed when a
corner of the polyhedron is sliced off) which is seen to be incorrect:
when Coxeter later co-authored [4] he gave the correct vertex figure
(Figure 5), for the polyhedron ^{5}/_{2} | 2 3.
These figures illustrate how *The 59* approached the icosahedra
as solids and was not concerned with internal structure. As a consequence
the term "face" changed in meaning: the great icosahedron was
originally named after the twenty triangular faces passing through its
interior, leading Schläfli to give it the symbol 3, ^{5}/_{2},
which does not follow if the term 'face' refers to the irregular external
regions of a solid form, as in **11 12**.

Figure 3 Face diagram 11 12 |
Figure 4 Vertex figure of 11 12 |

Figure 5 Vertex figure of ^{5}/_{2} | 2 3 |
Figure 6 Face diagram 7 9 10 |

Face diagram **7 9 10** consists of three disconnected
*m*-shaped groups of regions (Figure 6). It is not possible to tell
directly whether the centre of the stellation is solid or hollowed out: in
fact it is the regular compound of ten tetrahedra. The inclusion of
internal structure is forbidden by Rule (iv), which is now seen to result
in incorrect face diagrams.

Rule (ii) has a proviso that the parts of a face 'may be quite
disconnected.' This has no relevance to the main purpose of the rule,
which is to secure icosahedral symmetry. The proviso was added because,
with no consideration of internal structure possible, it would have been
unacceptable to exclude face diagrams such as **7 9 10**. As
a result, oddities such as **g _{1}** ,

Now we turn to the question as to whether any acceptable stellations
might have been missed from *The 59*. Consider **De _{1}**f

Figure 7

**De _{1}**f

Closely related to these are **De _{1}**f

We have seen that the original rules led to unacceptable consequences due to their failure to recognise the internal structure of a polyhedron, in turn a consequence of the approach to polyhedra as solids rather than collections of faces. Bearing in mind that the rules were originally proposed with no clear idea of the outcome, they are well overdue for revision.

Figure 8

**De _{1}**f

Adherence to rules (i) to (iii) is necessary to secure
icosahedral symmetry. We will want to be able to approach the icosahedra
as collections of faces, so we should make explicit a fundamental rule
which is assumed in *The 59* (which approached polyhedra as
finite solids), namely, 'The faces must close around a finite region
of space.' These four rules could probably be simplified using Group
theory.

Beyond this point answering the question, 'What are the defining
characteristics of a stellation?' is not much easier than it was
sixty years ago. I wrote a computer program to investigate some possible
new rules in place of (iv), (v) and the proviso in (ii). The investigation
was not systematic. Adherence to the four rules given above was achieved
by *ad hoc* coding as required at the time: the code has not been validated
beyond its ability to reproduce *The 59* from Miller's rules,
so its results can only be taken as indicative. While none of the new
rules proved entirely satisfactory they yielded some interesting insights.

Starting from the definition of a stellation given at the beginning of
this essay, I tried supplementing rules (iv) and (v) with 'All cells
must be connected to another,' which negates the proviso of (ii), and
'The central icosahedron must be present (for if it is absent, it
cannot be stellated).' These produced rather limited subsets of *The 59*.

Again from the definition of a stellation, we know that the properties of faces and edges are important. I suspected that the connectedness of the edges and/or faces is more fundamental that the connectedness of shells, so I tried substituting for rule (v), 'The faces must be continuous through the body of the icosahedron.' Although the total of 36 icosahedra allowed was still rather limiting, this found the two lost icosahedra described above and so confirmed the importance of the faces.

Next I investigated, 'All edges must be continuous through the body
of the icosahedron, i.e. no separated collinear segments are allowed.'
This again was less than a full answer while finding some interesting
icosahedra; two are discussed here. Each face of **acdf _{2}g_{1}**
(Figure 9) is bounded by a single 15-sided polygon. It has the same
external form as

Figure 9

**acdf _{2}g_{1}**

Figure 10

**be _{2}**

The obvious next step was to explore an equivalent continuity rule for
faces which would, in the language of *The 59*, be something
like, 'The bounding edges in any one plane must form a single circuit
or set of overlapping circuits about the centre.' In seeking to code
this rule I found I had to consider the density of certain regions,
especially the 'empty' inner regions, of the polygons and the fundamental
question arose, what is a polygon? More specifically, can a star polygon
be part 'filled-in' and part 'empty' and what rules apply - for example is
the face diagram in Figure 9 a polygon? For a star polygon the
density of a given region is the number of times the region is covered by
a ray *OP* joining the centre of the region *O* to a point
*P* as *P* makes a circuit round the edges of the polygon. The
region outside has density equal to 0 and all regions inside have
density greater than or equal to 1. For our purposes, we will think
of regions with density greater than or equal to 1 as filled-in, and
regions with density equal to 0 as empty. Referring to the star
polygon in Figure 5 again, the pentagonal region in the centre has
density equal to 2. But it also has density congruent to 0 (Mod 2),
i.e. counting the density modulo 2 the star would have an empty, or
hollow, centre. The density of the face diagrams in Figures 9 and 10
is also 1 modulo 2. Are these valid polygons? And is there any
value in counting the density modulo 3, 4 and so on? Modular arithmetic
would allow a more rigorous investigation of stellations whose faces are
partially filled-in polygons, but I have not come across any previous work
on such polygons: I would be grateful if any reader can point me to any
suitable references.

Once we have a satisfactory definition of a 'polygon' for our purposes, we can go on to re-examine the principle of stellation, which should in turn lead to a better set of rules and a new enumeration of the icosahedra (and stellations of other polyhedra, too). We will want to recognise which regions of face planes, in the interior of the polyhedron, are filled-in and which are empty. Finally, we may have to ask whether there is a single 'right' principle of stellation to be found or to what extent it might depend on one's interests at the time.

A face plane of the icosahedron is shown, giving the 18 lines of intersection with other face planes (one plane is parallel and does not intersect). The face of the original icosahedron is shown shaded. |

The illustrations were created with the help of !Stellate software for RISC OS computers, from Fortran Friends, PO Box 64, Didcot, Oxon. OX11 0TH.

The variations in Miller's rules were investigated on a computer using an *ad hoc* coded program. Its results cannot be guaranteed but I am not aware of any errors. The lists of stellations according to these variations are given a series of Program Printouts.

Since writing the above article, it has occurred to me that the rules for finding stellations will be paralleled by equivalent rules for the duals of the stellations.

Every polyhedron has a dual or reciprocal polyhedron (obtained by reciprocating the polyhedron with respect to a sphere). For example the vertices of a uniform polyhedron (e.g. an Archimedean polyhedron) are equidistant from its centre, while the faces of its dual are similarly equidistant. Certain rules will define the relationship between polyhedra within a whole family of shapes, and reciprocal rules will define the reciprocal relationship within the family of duals.

The rules governing the duals of the stellated icosahedra, and their reciprocal relationship to the rules governing the stellations themselves, need to be investigated in parallel with the stellation rules, as through duality they effectively become a single topic.

- H. S. M. Coxeter (et al),
*The fifty-nine icosahedra*, 3rd Edition, Tarquin (1999). - A. H. Wheeler, Certain forms of the icosahedron and a method for
deriving and designating higher polyhedra,
*Proc. Internat. Math. Congress, Toronto, 1924*, Vol. 1, pp 701-708. - P. Cromwell,
*Polyhedra*, CUP pbk. (1999). - H. S. M. Coxeter (et al), Uniform polyhedra,
*Phil. Trans.***246 A**(1954) pp. 401-450.

Guy Inchbald

guy@steelpillow.com

*Park View, Queenhill, Upton-upon-Severn, Worcs WR8 0RE, England*

Updated 24 May 2017