Icosahedral precursors

Introduction

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 very 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. facettings of the dodecahedron will sometimes be referred to as facetted dodecahedra or just dodecahedra, though most of them do not have exactly twelve faces.

A polyhedron is treated as an irreducible closed set of polygons, comprising two subsets (faces and vertex figures) whose interconnections are closed under recursion. A polyhedron may have a complicated internal structure. Some further refinements are discussed in the text.

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.

A compound polyhedron is a set of polyhedra sharing a common centre and disposed symmetrically - here, typically with icosahedral symmetry. As is usual, we distinguish between compound polyhedra and compound stellations or facettings as follows. We 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.

The principle of duality, by which every polygon or polyhedron has an opposite twin, is well known. We distinguish here between topological duals and geometric reciprocals. The reciprocal figure to a polyhedron is the geometric figure obtained when the polyhedron is reciprocated with respect to some sphere. Where the sphere and polyhedron are concentric, so also is the reciprocal. This may be called the standard dual, and is generally implied wherever the term "dual" is unqualified.

Precursors

I have previously noted the dual relationship between face diagrams of the stellated icosahedra and vertex figures of the facetted dodecahedra [1]. I also 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 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. In order to avoid confusion with the generators found in group theory, I have now renamed these as precursors. The same idea may be applied to polygons, such that the precursor defines a dual pair of polygons. It may also be extended to groups of polygons. 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.

We know that non-convex polyhedra can require that vertices, edges and faces all be specified. The great dodecahedron has the same edges and vertices as the icosahedron but different faces, the same face planes and vertices as the small stellated dodecahedron but different edges, and the same face planes and edges as the great stellated dodecahedron but different vertices. Only by specifying all three, viz. faces, edges and vertices, can the great dodecahedron be uniquely identified.

One convenient way of doing this is to define a polyhedron not only by its faces but also by its vertex figures. For example we can readily see that the various dodecahedra just mentioned each has a unique combination of face and vertex figure. In the present case, one type of polygon lies in the face planes of the icosahedron (the icosahedral precursors) and the other in the face planes of the dodecahedron (the dodecahedral precursors). Once one type is defined, be it face or vertex figure, the imposition of icosahedral symmetry forces the other. So for the present purpose it is sufficient to analyse only one set - the icosahedral precursors.

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 [1]. 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 p_d and p_l. 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).

Where multiple congruent circuits are made, the short form of the precursor describes one 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.

The tidy facets are listed in [1]. A tidy edge comprises exactly two sides. Sides which count only once per edge are joined to only one facet and so are open and available for joining to facets in other groups, those which count exactly twice are joined to two facets and so are closed edges within that group, and those which count more than twice are untidy.

Tidiness

The term "polyhedron" has been applied to a wide variety of three-dimensional forms. Nowadays, we distinguish between the the geometric properties and the topological, or combinatorial, propertices of a polyhedron. An abstract polyhedron is a partially-ordered set (poset), which must be realized by mapping onto some geometrical figure. The poset and its realization have the same topology.

Combinatorially, any polyhedron comprises a finite, irreducible closed set of polygons (faces), where any edge joins exactly two vertices and two faces. A tidy polyhedron is one whose geometry conforms to the more standard definitions. For example its faces must be flat and no elements (faces, edges, vertices) may coincide. Any given abstract polyhedron will have many possible realizations, some of which will appear to be tidy and others not.

Tidiness is not preserved under reciprocation: a geometric reciprocal of a tidy polyhedron will not necessarily be tidy. In the present context, two particluar 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 also accept polyhedra whose faces and edges extend to infinity. Such infinite polyhedra exist in extended Euclidean space. 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!). In passing, we may note that in elliptic spaces all such polyhedra are finite, which can sometimes be convenient.

This increased number of possibilities leads in turn to a great number of facettings and stellations; so great in fact that analysis and enumeration become formidable tasks. In passing, one may note that for hemi facettings with edges s present, the reciprocal icosahedra must have cells with edges s, that is k₁ and/or k₂.

No doubt many 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 rules based on the properties of faces, edges and/or vertices are doomed to failure - I suspect that the emergent nature of the whole polyhedron must be taken into account.

For example look at 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"). Comparing its face diagram and dual facetting with those of f₂, there appears to be little significant difference, yet f₂ is just a collection of disconnected fragments. Their precursors reveal no significant differences either, suggesting that the value of precursors is ultimately descriptive rather than analytical.

Stellation Af2

Face GnEpdDplEn~EplDmDpd

The analytical part of any solution must it seems be embedded in the rule set used to select valid precursors. So where [3] applied Miller's rules to faces of surhedra and to cells, the approach adopted here is to apply some modified rule set to precursors. The rules must be able to distinguish properties of the whole polyhedral form, i.e. they must be concerned with the emergent properties of the precursor set.

The need to consider the whole polyhedral form may not be too unexpected. Firstly, along with the recognition of internal structure go such ideas as false vertices and false edges - we want a system which bypasses such false phenomena. Precursors achieve this, so the rules for selecting precursors will need to be at an equal level. Also we know that non-convex polyhedra can require that vertices, edges and faces all be specified. For example the great dodecahedron has the same edges and vertices as the icosahedron but different faces, the same face planes and vertices as the small stellated dodecahedron but different edges, and the same face planes and edges as the great stellated dodecahedron but different vertices. Only by specifying all three, viz. faces, edges and vertices, can the great dodecahedron be uniquely identified.

Finally, when considering stellations and facettings, we must also add rules to distinguish acceptable compounds from the unacceptable. Compounds can create various unsatisfactory situations. Firstly, where the result is a compound of two atomic stellations/facettings. This may be dealt with by specifying that no subset of the solution should also be a solution. Secondly, where the result comprises a scatter of polyhedra around an empty centre - what may be generally termed a composite and in the present context a constellation. My first idea to deal with this was that at lease one circuit in any precursor must surround its centre; however this did not exclude all constellations. I found it necessary to ensure that all polyhedra surround the centre of symmetry. Thirdly, where the result may be deemed trivial. For example if a single corner of a dodecahedron is truncated, a compound of twenty will have the required symmetry but the outward appearance of a plain dodecahedron. Its dual is an icosahedron with a single face raised into a low pyramid, again a compound of twenty will have the required symmetry but the outward appearance of the first stellation, B. By most criteria, such results are trivial. We may note that the icosahedral precursor for this example compound includes 19 superimposed congruent (i.e. non-unique) circuits and three others. Taking the opportunity to simplify certain rules using Group theory, here is a possible rule set:

1. The precursors must comprise two recursively closed subsets.
   This ensures the formation of complete polyhedra.
2. Every circuit within a compound precursor must be unique.
   This (hopefully) eliminates trivial results.
3. The precursors must lie within the rotational symmetry group of the icosahedron.
   This ensures overall icosahedral symmetry, while allowing chiral forms.
4. One precursor subset must lie in the face planes of the icosahedron.
   Together with iii. this ensures we are dealing with stellations of the icosahedron and facettings of the dodecahedron. It should also force the location of the other subset.
5. Any discrete subsets of precursors must be congruent.
   This forbids compound stellations while allowing compound polyhedra. That is, it defines atomic stellations.
6. Any discrete subsets of precursors must be concentric.
   This forbids constellations.

The rules apply equally to the facetted dodecahedra and to the stellated icosahedra and, with appropriate modification of iii. and iv, to all polyhedra. Rules i. and ii. are new, and are required by the explicit need to consider internal structure. Together with iii. they are also purely combinatorial in character. Geometry is not referred to until Rule iv. The four rules iii. to vi. 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). It remains to be seen whether these rules are adequately concerned with the emergent properties of the precursor set.

I have previously [2] mentioned the difficulty of allowing for certain kinds of hole in a polygon or polyhedron. My rule vi. forbids holes in compounds, but not holes in individual polyhedra. Also, some facettings and stellations may best be seen as n-methoric or n-synaptic forms. A precursor essentially traces a circuit, which is not particularly useful in these contexts: holes punch through overlapping circuits or loops and only really become significant when viewed from a higher dimension, and n-synaptic edges form more complicated structures than simple linear circuits. It is not clear to me whether the idea of precursors can sensibly be extended to cover such cases, or even whether it should be.

Well, here I am, after many years and four papers, pretty much back to 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?

References