Home Stellating the icosahedron
Home Stellating the icosahedron
It's taking me a long time. The stellations of the regular icosahedron grabbed my attention ten years or so. 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 17th and 18th 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, the idea of a "face" changed subtly but unintentionally in meaning, greatly affecting the result.
Bridge enumerated only a fraction 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 evidently have duals which are 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, 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.
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 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 it 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 the “polyhedra” which we are talking about. 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 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, which does not really concern us here, is the way that modern topologists and combinatorialists have adopted 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.
And finally there can be a blindness to fundamental mathematical ideas such as continuity, or to the distinction between 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:
Only the last two 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). Hopefully, I will be able to chronicle my progress back up – watch this space. However, if you prefer to tread this path for yourself, here is some suggested reading – again, 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)
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.
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.