Home Stellating the icosahedron
Last updated 27 March 2011
27 March 2011: New topological discussion. Definition of a polyhedron refined. Reciprocity answered. References to Stewart, Lakatos. Minor corrections.
20 Dec 2010: It gets worse. The modern theory of abstract polytopes is at least an attempt to fix things up, but it also fails. Also I have updated my own definition of a polyhedron.
It's taking me a long time. The stellations of the regular icosahedron grabbed my attention ten years or so ago. Stellation theory seemed to have been pretty much wrapped up by then, with the last word being spoken in 1938 when Coxeter, DuVal, Flather and Petrie published “The fifty-nine icosahedra”. But I was dissatisfied, things didn't seem quite right. I set out to get to the bottom of things and to provide a more useful enumeration – it seemed straightforward enough, a nice little project. But what a journey it has turned out to be! My pages on stellation and facetting flourished for a while and then went quiet. But I am still working on it, and hope eventually to come up with the goods. In the meantime, this note tries to explain what is going on.
The ancient Greeks started it all. By the time of Plato the five convex regular solids had been discovered, and Euclid went on to synthesise the geometry of space. Star and other non-convex polyhedra become popular during the Renaissance, leading eventually to Kepler's idea of stellation and his discovery of regular stars. A great explosion of synthetic geometry during the 18th and 19th Centuries, mainly in France and Germany, led to Cayley's understanding of the polar or dual relationships between the regular stars, and of their densities. At about the same time, Schläfli discovered higher-dimensional analogues of polygons and polyhedra which Stott later dubbed polytopes; polygons and polyhedra could now be understood as two- and three- dimensional examples of a more general picture. Along the way various stellations of the regular bodies had been found, but it was not until JCP Miller proposed his set of rules that Coxeter and DuVal could enumerate those of the icosahedron. The final coda seemed to have been written when Bridge, aware of the dual relationship between the processes of stellation and facetting, enumerated the facettings of the regular dodecahedron.
As I traced this process laboriously back through the archives, I became more and more astonished. Only the ideas of non-convexity and higher dimensions stood up to rigorous scrutiny. Every one of the rest is at best half-baked and incomplete, and some are fatally flawed.
Miller's rules turn out to bear little relation to the Keplerian idea of stellation, rather they are based on ideas of spatial decomposition developed by Wheeler and others. They jump into the narrative of “The fifty-nine icosahedra” as if from nowhere. Worse, they deliberately forbid any idea of internal structure and the idea of a "face" was changed subtly in meaning, greatly affecting the result.
Bridge enumerated only a handful of the possible facettings of the dodecahedron, confining himself to what we might call a tidiness of form. He even rejected certain facettings because the dual icosahedra were not tidy enough for him. Many of the fifty-nine icosahedra have duals which were evidently not tidy enough for Bridge.
The question, “What is a polyhedron?” has never been fully answered. Definitions have ranged from solids to surfaces to skeleta to combinatorial point sets, to realisations of abstract posets (abstract polytopes) with all sors of features such as infinite extent, coincident elements and so on allowed by some investigators but not by others. Today the debate rages perhaps stronger than ever. Particularly unfortunate is the habit of defining a polyhedron as a solid, then later describing tilings of the plane as "infinite polyhedra", remarking that they differ only in not having a solid interior. This boils down to the logical absurdity that "a tiling is a solid which is not a solid": of course what has actually happened is an unconscious slip from polyhedra as solids to polyhedra as surfaces. Another vexed confusion arises over whether the edges of a polyhedron are finite segments or infinite lines and whether or not it matters.
Our ideas of inside and outside are clouded by the popularity of two different and incompatible approaches; the one based on the idea of "outside" as containing infinitely many straight lines, and the other on the idea of the surface wrapping round a dense interior. The densities determined for regions within self-intersecting polygons and polyhedra depend on which of these approaches we take, whether the surface is orientable or not, how we choose to interpret a density of zero, and even what kind of space we placed our polyhedra in to start with.
The synthesis of geometry leads first not to Euclidean space but to general projective space. This latter space is the home of polarity, and in general the polar or reciprocal of some polyhedron is not at all what we have been led to believe: polyhedral duals are evidently something else.
I am not alone in seeking some way through the wider mess that is polyhedron theory today, and along with other investigators have found various obscuring principles at work.
One class of problem comes from our failure to define the things we are talking about. From the days of Euclid we have consistently failed to define a “polyhedron”. Other ideas, such as their reciprocity or the kind of space we are putting them in, have since fallen into the same trap.
A related class of problem comes from a habit of twisting some well-understood term to a new meaning inconsistent with the old. Miller's rules are a prime example. Another is the slip from solids to surfaces. Yet another, which does not really concern us here, is the way that modern analytical disciplines have adopted various definitions of polyhedra and polytopes which are quite alien to their origins in pure geometry.
Many principles first enunciated in ages past have become detached from their original context, enshrined in mathematical folklore and applied willy-nilly out of context. Examples include the assumption of Euclidean space, and the idea that every polyhedron has a dual which may be found by reciprocating it about a concentric sphere.
That last is also an example of the more general fault of unjustified generalisation from the particular. Prime examples here include taking results from the study of convex, symmetrical polyhedra and assuming that they apply also to non-convex or asymmetric polyhedra such as star pyramids. For the most part, they do not.
Then there is the recent habit of concept transfer - providing some rather abstract but at least rigorous algebraic definition, declaring a geometric polyhedron to be a "realisation" of the abstract form, and assuming that the rigour of the abstract concept has solved all the problems. However at best this merely transfers all the old questions onto "what is a realisation?" Worse, it can disguise the fact that some of these abstract forms may not be want we want anyway - for example the theory of abstract polytopes allows figures which are inconsistent with the use of Euler's formula in the characterisation of polytopes and therefore have no definite topology.
And finally there can be a blindness to fundamental mathematical ideas such as continuity, or to the distinction between regular polytopes and configurations (seen in that between the quadrilateral polygon and the complete quadrilateral and complete tetragon configurations).
The whole thing is utterly shambolic and I am, as I said, astonished that this sorry state of affairs has been allowed to develop. You can perhaps now understand why it has taken me such a long time to get to the bottom of things, especially as I am not a professional mathematician in any way. And now that I have at last found the starting point, I can begin to build back up towards something that might hold together a little better and perhaps even stand the test of time.
Here are nine questions that need answering, with each tending to build on the previous ones. I have now found answers to most of them, which I hope to expand on in due course.
The most fundamental kinds are incidence spaces, which may be discrete or continuous, finite or infinite: we can do quite a bit of geometry in these. Beyond these are morphic spaces such as projective and affine spaces, and beyond these are concrete metric geometries such as Euclidean, spherical, inversive, hyperbolic and many others.
Incidence spaces are too simple to be able to distinguish polyhedra from other constructions such as configurations. All the other kinds mentioned above are suitable, with projective space being the best starting point.
A polyhedron is basically a lump of stuff whose surface is divided into faces, edges and corner points (vertices). The "stuff" is best thought of as magic rubber or magic clay - it can pass through itself to allow star surfaces. More formally, a morphic polyhedron is a closed 3-manifold (solid) bounded by a connected piecewise surface (2-manifold). "Closed" here means that the 3-manifold includes its boundary. A polygon is a closed 2-manifold bounded by a circuit of line segments and points, and a polytope in p dimensions is a closed p-manifold bounded by a connected piecewise manifold of (p−1) dimensions. Typically we will want to map a polyhedron into ordinary 3-space so that we can see it, although that is not strictly necessary. If our main interest lies with traditional (flat-faced or epipedal) polyhedra in 3-space, we must be explicit that we are confining ourselves to this variety.
The inside is just the bounded 3-manifold (without its boundary). When it is embedded in ordinary space, it may be twisted up to create a non-convex figure such as a star. Interestingly, we can choose different manifolds to "fill" inside a given boundary. The resulting star may or may not obey the usual "density" rule, depending on what manifold we choose and how it is then twisted up.
No. In a piecewise decomposition or partitioning, all faces must be topological discs, so such holes are impossible. However, it can have "false" holes where the body wraps around some part of empty space.
It is broadly similar to projective polarity, except that the dual edge is that segment, of some line, which does not cross infinity (and likewise for faces). Commonly, the polyhedron is reciprocated about a concentric sphere; the result may be called the standard dual of that polyhedron.
Yes and no. Reciprocation is a purely geometrical operation and can result in degenerate figures that comprise multiple images of part or all of the polyhedron. These composite images have the correct morphic structure but the geometrical mapping is degenerate: such a figure is a "morphic" polyhedron but not a geometrical one.
Only the last two questions deal with stellation – the rest are all to do with the general theory of polyhedra. The starting point goes right back to the foundations of geometry itself (yes, polyhedron theory really is that broken).
Now at last, in 2011, I can at least outline my approach. When Euler discovered his famous polyhedron formula V − E + F = 2 he founded a discipline he called stereometry, the analytic study of polyhedral and spatial structures. Poincaré later developed the field hugely and called it analysis situ – the analysis of position. In the 20th Century the discipline became known as topology and came to lie at the heart of many mathematical disciplines, such as the Calabi-Yau manifolds of string theory in modern physics. Much of the 20th Century was spent proving Poincaré's ideas - the famous conjecture that bears his name was not proved by Perelman until the dawn of the 21st century. One of Poincaré's chief tools was the decomposition of a topological object or manifold into polygons, polyhedra and so forth, maintaining the strong link created by Euler between the disciplines. In the latter half of the 20th Century Stewart studied toroidal polyhedra, defining his figures as topological surfaces. The theoretical significance of his approach seems to have been lost on his mainstream contemporaries, who were all busy developing more fashionable set-theoretic ideas (at that time set theory was touted as the universal foundation of all mathematics and even philosophical logic. It has since failed to live up to expectations). Essentially I take Stewart's approach another step back towards Poincaré's, by considering not only the surface but also the interior as a topological object. This is something that the set-theoretic approach is quite unsuited to address.
Remarkably, this has led me to the discovery of a whole new class of regular star polytopes, including various new regular dodecahedra and icosahedra. Many if not all of these regular stars are valid stellations, though universally forbidden by Miller's unfortunate rules. You see, I'm only trying to stellate the icosahedron the way it should be stellated, and look where the journey has been taking me!
I hope to properly chronicle all this and more in due course – watch this space. Meanwhile if you prefer to tread the path for yourself, here is some suggested reading, broadly in order of progress from the ground up:
Coxeter, HSM; Projective geometry, 2nd ed, Springer Verlag (1974)
Greenberg, MJ; Euclidean and non-Euclidean geometries, 2nd ed, WH Freeman (1980)
Hilbert, D & Cohn-Vossen, S; Geometry and the imagination, 2nd ed, Chelsea (1999)
Richeson, D; Euler's Gem - The polyhedron formula and the birth of topology, Princeton (2010)
Lakatos, I; Proofs and refutations - The logic of mathematical discovery, CUP (1976)
Grünbaum, B; Are your polyhedra the same as my polyhedra? Discrete and comput. Geom.: the Goodman-Pollack Festschrift, ed. B. Aronov et al, Springer (2003), pp. 461-488.
Stewart, B; Adventures among the toroids, self-published (1970).
Grünbaum, B & Shephard, G; Duality of polyhedra, Shaping space – a polyhedral approach, ed. Senechal and Fleck, Birkhäuser (1988), pp. 205-211
Gailiunas, P & J. Sharp, J; Duality of polyhedra, Internat. journ. of math. ed. in science and technology, 36, No. 6 (2005), pp. 617-642.
Wenninger, M; Dual models, CUP, 1983
Bridge, NJ; Facetting the dodecahedron, Acta Crystallographica A30 [1974), pp. 548-552.
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