Updated 14 Dec 2016
This essay is unfinished and represents a work in hand.
Precursors provide a way to describe and identify the stellations of the regular icosahedron and their reciprocal facettings of the dodecahedron. Some theoretical foundations are presented. Some problems with untidy polyhedra are discussed, and yet another lost stellation noted. Revised rules for identifying stellations and facettings based on precursors are proposed.
14 Dec 2016. Morphic realizations, general tightening up, revised rule set.
21 May 2005. Renamed from "Icosahedral templates", to reflect another change in terminology. Some useful clarifications, including a slight reordering of my new rules and updated commentary. Still awaiting more general theoretical work.
This essay presents the idea of precursors as applied to the faces of an isohedral polyhedron and the vertex figures of its (isogonal) dual, in the context of my continuing analysis of the stellations of the regular icosahedron and their dual facettings of the regular dodecahedron. It is the fourth in a series of essays contributing to this analysis. It remains unfinished and records some of my current work in hand.
First, I should define some of the terms I will be using. References to the icosahedron and dodecahedron are of course to the regular varieties.
A polyhedron is treated as a closed surface or manifold in 3-space, being comprised of (or decomposed into) elements which include flat faces, straight edges and point vertices. A polyhedron is allowed to fold back and intersect itself to create an internal structure.
A facet of a polyhedron is a polygon whose vertices are also vertices of the polyhedron; it may be located partially or fully within the interior. Trivially, all faces are facets.
A compound polyhedron is a set of polyhedra sharing a common centre and typically disposed symmetrically. As is usual, I distinguish between compound polyhedra and compound stellations or facettings as follows: I accept as atomic stellations or facettings, just those compounds where the polyhedra are congruent. Those with non-congruent polyhedra are treated as non-atomic compound stellations or facettings having no interest within the present context.
It will sometimes be helpful to distinguish between the vertices and edges of a polyhedron on the one hand, and the corners and sides of a polygon on the other. Thus the vertex figure of a polygon is referred to as its corner figure. A vertex figure may be thought of as the surface exposed when a vertex of the polyhedron is cut off in a certain way. A corner figure is likewise the line exposed when a corner of the polygon is cut off in a certain way. Where corners have differing angles, the corner figures are characterised by their various lengths.
I have previously noted the dual relationship between face diagrams of the stellated icosahedra and vertex figures of the facetted dodecahedra . I labelled these figures in such a way that the same letter sequence may be used to represent both a given face diagram and (with different capitalization) its dual vertex figure.
The face diagram of a given stellation may be represented by an anticlockwise circuit around the figure, noting the sequence of vertices and edges traversed. The face of a stellation is thus uniquely identified by a sequence of letters. For example the icosahedron has vertices A and edges m, so its face is represented by AmAmAm.
The vertex figure of a facetting may be represented by a similar circuit, noting the sequence of face sections [corner figures] and edge sections traversed. A face section [corner figure] is a side in the vertex figure, and an edge section is a corner in the figure. A similar sequence of letters may be used. Because the face diagram and vertex figure are reciprocal, it is convenient to label them in such a way that the same letter is applied to the reciprocal features of each. For example point A in one diagram reciprocates to line a in the other. We find that reciprocal circuits generate the same sequence of letters, but with differing capitalisation.
For example the dodecahedron itself has faces A and edges m, its vertex figure shows sides (face sections) a and corners (edge sections) M, and so is represented as aMaMaM. We may call this the dual, or reciprocal, capitalisation to the icosahedron's face AmAmAm.
All the figures here have trigonal symmetry, so we may economise by recording only the first third of the circuit, the remaining two thirds being exact repeats. The icosahedron is then just Am and the dodecahedron aM.
I have adopted the convention that the circuit around a face starts at a vertex, whereas that around a vertex figure starts at a face (i.e. corner figure). The two dual sequences can thus be distinguished in that the face of a stellation starts with a capital letter, while the vertex figure of a facetting starts with a lower-case letter.
I also proposed that a single mathematical representation, or generator, might be used to define both a polyhedron and its dual. The same idea applies to polygons, in that a dual pair of polygons, such as a face of one polyhedron and a vertex figure of its dual, are defined by the same underlying generator. In order to avoid confusion with the generators found in group theory, I have here renamed such a common underlying object as a precursor. In the present case the letter sequence, without any capitalisation, may be used to represent the precursor of a dual pair of face diagram and vertex figure. Collectively, these may be called the icosahedral precursors after their associated symmetry group.
The facet planes of the dodecahedron may be identified as A to K working from the outside in, similarly the edge sets as m to s. These are illustrated in . For certain planes, different groups of edges can be chosen to give different facets. For example in A, we may choose either edges m to form the sides of a convex pentagon, or edges n to form the sides of a pentagram. These may be represented as Am and An respectively. Where edges are chiral, as p, they may be divided into dextro and laevo sets, as pd and pl. The chiral edge sets are dual to each other. Chirality is a geometric phenomenon, not a topological one, so the dextro and laevo sets must be topologically distinct in some way. From their duality, we can deduce that they lie in pairs, with each member at right angles to its oppposite-handed partner (in this light, the compound of ten tetrahedra may be seen as a compound of five stella octangulae).For some faces, such as C, only certain of the edges in a set may be used. Where such facets are filled such that they lie in the wider angle between the selected edges, giving a more "full" appearance to the compound, the arrangement of edges is said to be "ventral," as nv. Where the facets are filled such that they lie in the narrower angle between the edges, giving a more "cutaway" appearance to the compound, the choice of edges is said to be "retro," as nr.
Where multiple congruent circuits in the plane are made, a short form of the precursor notation describes one such complete circuit. Where multiple non-congruent circuits are made, their individual precursors are separated by a ~ (tilde) symbol. A set of such circuits may be called a compound precursor and uniquely defines a dual pair of polygon sets. A compound precursor may be shortened as before.
In the modern theory of abstract polytopes, the set-theoretic partial ordering of its various elements is distinguished from its geometrical appearance. The physical form is said to be a realization of the abstract set. The set of abstract elements is organized in ranks representing dimensionality. The relation of incidence is established between a pair of elements, each from adjacent ranks. Certain rules govern the assignment of incidences, such that the partially-ordered set captures the general structure of the polyhedron in terms of faces, edges and vertices. A polyhedron is realized by mapping the abstract elements into a real geometric space, such that incidences are preserved with their usual meaning.
Crucially for the present purpose, the order in which the ranks are realized is not unique. If the ranking order is reversed in the realization, the resulting geometric figure is the dual of the first. The applicability to precursor theory becomes obvious.
I regard realization as a two-stage process. The first step is to decide how each rank is to be realized - as vertex points, edge segments, face regions and, if you are interested in the interior, a region of space. Reverse the choice and you will obtain the dual polyhedron. With apologies to Hilbert's apocryphal remark, you could instead realize the abstract form as tables, chairs and beer mugs, so this step is not trivial. What this first step in realization creates can be thought of as a kind of stretchy rubber quasi-figure without any ideas of size or flatness. I call this a morphic polyhedron.
The field of topology is sometimes called rubber-sheet geometry because it deals with the general properties of the form rather than the exact geometry. A morphic polyhedron is technically a certain surface decomposition of the associated rubber sheet or manifold.
A precursor must be defined at the abstract stage, as by the time it reaches the first, morphic stage of realization it has already taken one or other dual form - as a face of the one dual or a vertex figure of the other. Formally, a precursor is a set of certain congruent sections of the abstract figure, where congruence is defined in terms of the symmetries of the abstract set and a section has a specific meaning in abstract theory.
It is convenient to define a "face section" as the face itself together with the incident edges and vertices, along with a "vertex section" as the vertex itself together with the incident edges and faces. These sections have a dual character, in that a face section of one polyhedron is also a vertex section of the dual polyhedron: one section, two dual realizations. This is of course exactly the property for precursors which we seek.
A compound precursor is a set of one or more such sets of congruent sections. However there is no way in which an abstract or morphic figure can be analysed to reveal which such sets we might wish to make coplanar and hence form a compound precursor: that is entirely down to the second stage of realization.
The second step in realization is to map or inject this morphic figure into real space and tidy it up, as it were, constructing it as a conventional figure with plane faces, sraight edges and - in the present case - appropriate geometrical symmetry. Exact size is another such attribute but it need not concern us here.
In "rubber-sheet geometry" we are dealing with a magic rubber which can pass through itself, deform indefinitely and even shrink to a point. This allows any given morphic figure to have multiple realizations with the required symmetry. For example the great icosahedron has the same morphic form as the convex or Platonic icosahedron, while the great stellated dodecahedron has the same morphic form as the convex or Platonic dodecahedron. In passing we may note that consequently all four of these polyhedra, existing as they do in dual pairs, share the same abstract structure.
A key idea with such star polyhedra is that of internal structure - parts of the figure continue on inside it, even though we cannot see them. In The Fifty-Nine Icosahedra, the deliberate exclusion of this property from Miller's rules is without doubt its greatest failing.
A more difficult aspect of the final realization occurs where we create a compound figure comprising multiple self-contained polyhedra. Abstractly, there is nothing to choose between say the regular compound of ten tetrahedra and the "constellation" of twelve pentagonal trapezohedra, stellation f2. Even at the morphic stage there is no concept of where each polyhedron lies in relation to the others. this only comes when we set out the face planes and symmetries.
Similarly, a compound precursor is not formed until this stage, as up until now there has been no plane defined for it to lie in. Nor can the precursor notation be applied until this stage, as before then it is not evident which edges may end up being collinear or even coincident.
There are other difficulties with the precursor notation. Where a line of intersection divides into multiple segments such that they fall into multiple symmetry groups, these groups may only be distinguished by the adjacent letters indicating the end points of the segments. Once capitalised, the notation describes not so much a precursor as an actual polygon group. Provided we are aware of these issues, they need not trouble us here. For convenience, I will continue to describe such groups as precursors (much as one continues to describe a realization, with its faces vs. vertices concretised, as a polyhedron).
Sometimes a geometrical figure appears to be more or less a polyhedron but is degenerate in some way. Perhaps its structure overlaps in a visual muddle, perhaps it is squashed flat into the plane or has curved sides or whatever. The trouble is, nobody can agree on what is acceptable and what is degenerate. What is positively desirable to one mathematician may be unacceptable to another. If one face can overlap itself with a density greater than 1, as in the stellated dodecahedra, why should not two faces overlie each other? Why not allow a side of zero length with its two vertices coincident? In this context I have used the idea of tidiness to try and capture our inutitive notions of what is acceptable. Whether such an intuitive tidiness can be formalised is an open question, it may be that out intuitions are irrational and inconsistent.
In conventional geometry and especially in the context of stellation and facetting, the most obvious requirement for tidiness is that the faces be flat, edges straight and the surface closed. These are inherent in the definitions of stellation and facetting themselves and so for the present analysis they may be assumed.
More significantly, I have suggested that a tidy figure may have overlapping elements but not coincident ones. On this basis, the tidy facets and facettings of the dodecahedron were listed in .
Reciprocation is a geometrical procedure which transforms a polyhedron into a dual figure. Where a polyhedron is reciprocated about a concentric sphere, I call the resulting figure its standard dual. Tidiness is not preserved under reciprocation: a reciprocal of a tidy polyhedron will not necessarily be tidy. In the present context, two particular problems arise.
A "hemi" polyhedron has one or more faces passing through its centre. In the standard dual, such faces reciprocate to points at infinity, so the dual extends to infinity. Such hemi forms are well established and accepted in the literature, so we will accept here those which are tidy in all other respects; they may be termed hemitidy polyhedra. To preserve the duality of stellations and facettings, we must accept duals whose faces and edges extend to infinity. Such infinite polyhedra exist in extended Euclidean space, which is the usual Euclidean space with an extra plane "at infinity", and are more naturally homed in projective space, where the plane at infinity is not intrinsically different from any other plane. By a natural extension of this argument, we must also accept edges and even faces lying wholly in the plane at infinity (provided they do not all do so, for then the figure would be a plane tiling and not a polyhedron). For hemi facettings of the dodecahedron with edges s present, the reciprocal icosahedra must have cells with edges s, that is k1 and/or k2. These cells define thick, flat, board-like slices across space.
The underlying issue is that a perfectly acceptable abstract polyhedron has been realized in a bizarre way, with some elements being mapped into space twice. In abstract theory the realization is said to be unfaithful. Whether one accepts these figures must depend on one's area of interest; here, they are accepted in order to preserve the principle of duality.
The second problem concerns coincident elements, such as vertices and edges. Coplanar faces reciprocate to coincident vertices, while collinear edges attached to coplanar pairs of faces reciprocate to coincident edges. Such duals appear untidy. They can also appear ambiguous, in that several different topological forms can have the same appearance. Consider a regular pentagon having vertices in the set V, edges in the set m, and with the diagonals drawn in, in the set n. The figure may be seen as a compound of the penatgon and a pentagram, with precursor Vm~Vn, or as a decagon VmVn, or a compound of five triangles VmVnVn, and so on. Many of these will appear as trivial repetitions of some valid figure, for example the compound of five trapezoids VmVmVmVn has many coincident m edges, and its dual is no better. But many others will have visibly distinct duals, and we will want to treat them as valid figures. As a result we find that many more types of face can be identified, especially in groups A, F and G.
In both these examples, tidiness has been sacrificed to mathematical rigour and the difficulties buried inside that mysterious second step in the realization process.
The increased number of possibilities introduced by multiple coincidences leads in turn to a great number of identical-looking facettings and stellations which differ only in the pattern of overlaps. In fact the repetitive overlaps rapidly become trivial and their number infinite. Identification and enumeration of the significant remainder becomes a critical task.
No doubt many borderline possibilities will turn out to be trivial, but which ones and why? This is the next major task ahead. Among the examples discussed here and earlier, one can find apparently acceptable ones where faces or vertices are visited by multiple circuits, or edges visited twice by the same circuit or by different circuits. Such examples suggest to me that any rules based on the properties of faces, edges and/or vertices are as doomed to failure as were Coxeter's segments or du Val's cells. I suspect that the emergent nature of the whole polyhedron must be taken into account.
Face GnEpdDplEn~EplDmDpd of Af2
For example consider Af2. Its face diagram shows the vertices where cells A and f2 touch to be false; it is a single polyhedron - and another lost icosahedron (perhaps both suprisingly and a little ironically, it also justifies the proviso in Miller's rule (ii) that the parts lying in a given face plane "may be quite disconnected"). In comparison, f2 is just a collection of disconnected fragments. The reason why one face diagram/precursor set leads to a single polyhedron and the other to twelve is not immediately obvious. In other cases we may find say twelve polyhedra which interpenetrate to fill a contiguous region of space – what are we to make of that?
Face GnEpdDplEn~EplDmAmDpd of Af2
The formative part of any solution must it seems be embedded in the criteria used to select valid precursors. So where  applied Miller's rules to visible regions of faces and to cells, the approach here is to apply some modified rule set to precursors.
One issue not yet developed is that, where the precursor notation identifies the whole of any edge between vertices of the parent polyhedron, in practice only a segment of that edge will be incorporated in the actual figure. In some cases two disjoint segments along a line may form edges of different faces, making it difficult to define structural criteria for a closed surface based solely on the precursor notation.
Fortunately here, Miller's rules make it an unspoken assumption that we are dealing with solids or closed surfaces and there is no rule to ensure this. We are therefore at liberty to do the same. However I find this lack curious as Coxeter independently introduced it into his analysis, so such a rule might be prudent. In this context I define a polyhedral set of precursors as one which forms a topologically closed surface (in set theory, "closed" means somthing rather different). A polyhedral subset is a subset of precursors, or parts of compound precursors, which forms a closed surface.
On the other hand Miller expended three rules merely to secure icosahedral symmetry, a requirement which can surely be simplified.
Another key difference is the need here to express the rules for precursors in terms which apply equally to stellations and facettings. While the nature of partial overlaps may be ignored, as they are not preserved under reciprocation, a set of satellites with no central core, such as stellation f2, should be avoided.
Based on such considerations, I propose a modified rule set:
The rules apply equally, by reciprocation, to the facetted dodecahedra as well as the stellated icosahedra and, with appropriate modification of ii. and iii, to all polyhedra. Rules i. and vi. are new. The four rules ii. to v. may be directly compared with Miller's five; besides being fewer in number, an additional advantage is that there are no qualifying provisos and no ambiguities over connectivity (both of which indicate muddled thinking on Miller's part). I have previously  mentioned the difficulty of allowing for certain kinds of hole in a polygon or polyhedron. My rule vi. forbids central holes in compounds, but not holes in individual polyhedra. It remains to be seen whether these rules are adequately concerned with the emergent properties of the precursor set.
Well, here I am, after many years, a new theory of morphic polytopes (yet to be published) and four essays, still pretty much back at where Coxeter et al. were when they started out - presented with some untried rules and no clear idea of their outcome. All I need to do now is to repeat their achievement by developing and working through a method of applying my new rules. Any willing helpers out there?