or, The Sorry State of Polyhedron Theory Today

24 Jan 2023: More on duality etc, second half rearranged.

18 Nov 2022: A few more details.

9 May 2022: Theoretical reappraisal essentially complete.

13 Apr 2021: Minor revisions.

It's taking me a long time. The stellations of the regular icosahedron caught my imagination in the late 1990s. Stellation theory seemed to have been pretty much wrapped up by then, with the last word being spoken in 1938 when Coxeter, Du Val, Flather and Petrie published *The Fifty-Nine Icosahedra*. But on close reading of their work I was dissatisfied, their results 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! I published a couple of essays, and my web pages on stellation and facetting continued to flourish for a while but then went quiet. Recently they have begun to pick up again, as a way forward is at last beginning to shape itself. In the meantime, this note tries to explain what has been going on.

The ancient Greeks started it all. By the time of Plato the five convex regular solids had been discovered. Euclid went on to synthesise the geometry of space, constructing Plato's polyhedra as his crowning glory. Star and other non-convex polyhedra become popular during the Italian 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 theory of polytopes in any number of dimensions. Along the way various stellations of the regular bodies had been found and the dual relationship between the processes of stellation and facetting discovered. But it was not until JCP Miller proposed his set of rules that Coxeter and Du Val could enumerate those of the icosahedron. The final coda seemed to have been written when Bridge enumerated the facettings of the regular dodecahedron.

As I traced these developments 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, or extending edges and faces. Rather, they are based on ideas of spatial decomposition developed centuries later 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 concept 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 him either. Even so, in some cases he found that several distinct stellations of the icosahedron all had duals which shared the same outward form and differed only in their internal structure. Bridge's total fell far short of the 59 one would expect of the duals of Coxeter's list. The idea that every stellation of one polyhedron has a dual facetting of the dual polyhedron evidently leads to some very unsatisfactory constructions.

Meanwhile Wenninger observed that dualising the "hemi" uniform polyhedra created infinite stellation-like forms which appeared not to be polyhedra at all. His result is inconsistent with the idea that every polyhedron has a dual polyhedron. But then again, different definitions of polyhedral duality may be found. The most common embodies the idea of polar reciprocity about a concentric sphere, yet this was the very one that Wenninger broke.

This further begs the question "What is a polyhedron?" It has never been properly answered, though it has been asked often enough. Definitions have ranged from solids to surfaces to skeleta to combinatorial point sets, to "realisations" of partially-ordered sets (abstract polytopes), while all sorts of features such as infinite extent, coincident elements and so on were allowed by some investigators but not by others. Some definitions even contrast a polyhedron of any number of dimensions with a polytope of the same number, the difference (on which they cannot agree among themselves) being the relative treatment of the bounding surface. Today the debate rages perhaps stronger than ever (e.g. Abrams & Elkind, 2019). Grünbaum remarked despairingly that a "polyhedron" means whatever you want it to mean. 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 being solids. 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.

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 or flat planes, and the other on the idea of the surface wrapping round a dense interior. The idea works well enough for convex polytopes, but only up to a point in non-convex (re-entrant or self-intersecting) cases. 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 polyhedron in to start with.

The synthesis of geometry leads first not to Euclidean space but to general projective space. Here there is as yet no infinity, no parallel lines or faces, no sense of size or angle. This latter space is famed for the occurrence of duality as a deep theorem, a theorem which Euclidean parallelism breaks. It is also the home of polarity, and in general the projective 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 mess that is polyhedron theory today, and along with other investigators have noticed various obscuring principles at work.

One class of problem comes from our failure to define the things we are talking about with adequate rigour. From the days of Euclid we have consistently failed to define a "polyhedron". Other ideas, such as their interior, 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 way that modern analytical disciplines have adopted various definitions of polyhedra and polytopes which are quite alien to their origins in pure geometry and often inconsistent with each other.

Many principles first enunciated in ages past have become detached from their original context, in which they are perfectly valid, and enshrined in mathematical folklore. They then get applied willy-nilly out of context. The slip from solids to surfaces is one example. Others include: various of those specialised definitions being applied in another discipline, 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 habit of concept transfer (e.g. Lakatos 1976). A modern example is seen in providing some rather abstract but at least rigorous algebraic definition, then 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 what we want anyway – the theory of abstract polytopes allows figures which are inconsistent with Euler's and Poincaré's approaches and therefore have no definite surface topology.

And finally there can be a blindness to fundamental mathematical ideas such as continuity (a particular preoccupation of Grünbaum's), or to the distinction between regular polytopes and configurations (seen in that between the quadrilateral polygon and the complete quadrilateral and complete tetragon configurations) and even what one might call a meta-blindness as to whether such blind spots matter (Coxeter was wilfully prone to this last).

The whole thing is utterly shambolic and I am, as I said, astonished that this sorry state of affairs has been allowed to develop. The starting point goes right back to the foundations of geometry itself. As I hope I have just shown, polyhedron theory really is that broken. The first thing I needed to do was to fully understand and define the problem.

One key principle I soon unpicked was the need to respect the idea that a polyhedron is a three-dimensional example of the more general polytope in any number of dimensions. Not every definition does so, but those which do not are related to higher mathematical ends and only incidentally find it useful to involve polytopes and polyhedra. In other words, such specialised definitions are not fundamental to the understanding of polyhedra as such, and so I have discounted them. Although I seldom mention higher (and lower) dimensional polytopes here, I maintain a careful consistency with them throughout. And sometimes they are crucial in exposing fallacies which are not evident in three dimensions.

I came up with nine questions that needed answering, with each tending to build on the previous ones. I have now found answers to all but the last of them, and am writing it all up bit by bit elsewehere. A summary to date is given below.

- What kinds of space can we do geometry in?
- Of these, which might be suitable for constructing polyhedra?
- What exactly is a polyhedron? And for that matter, what is a polygon or any polytope?
- What are the "inside" and "outside" of a polyhedron?
- Can a polyhedron have holes that are not topologically toroidal?
- What exactly are polyhedral duality and reciprocation?
- Can we now say that any figure reciprocal to some polyhedron is also a polyhedron?
- What exactly are the (reciprocal) processes of stellation and facetting?
- Which are the usefully distinct stellations and facettings of the regular polyhedra?

The most basic kinds are incidence spaces, which may be discrete or continuous, finite or infinite. We can do quite a bit of geometry in these, for example we can construct certain partially-ordered sets called abstract polyhedra (or, more generally, abstract polytopes). Beyond these are what I call morphic spaces, which have a smooth continuity but no idea of absolute measurement, such as projective and affine spaces, and beyond these are concrete metric geometries with measurable lengths and angles, such as Euclidean, spherical, hyperbolic, elliptic 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 piecewise into faces, edges and vertices (2-, 1- and 0-pieces). The "stuff" is best thought of as magic rubber – not only can it stretch and shrink but it can pass through itself to allow star surfaces. Crucially, I follow abstract polytope theory and allow the interior of a general piece to be non-simple. This relaxes the usual assumption that a polygonal face is necessarily disc-like, for example a star face may be a Möbius band. This has major consequences, and to fully define a given polytope the interior topology of each piece must be defined. I call these figures morphic polyhedra and, in general, morphic polytopes. Typically we will want to map such a polyhedron into ordinary Euclidean 3-space so that we can see it, although that is not strictly necessary. If our main interest lies with traditional (flat, disc-like faced or epipedal) polyhedra in 3-space, we must be explicit that we are confining ourselves to this variety.

The inside is just the lump of magic rubber (contained within its outer surface). When it is placed in ordinary space, it may be twisted up to create a non-convex figure such as a star. Because we can choose different manifolds (such as a Möbius band) to "fill" inside a given boundary, the resulting star may or may not obey the usual "density" rule, depending on which manifold we choose and how it is then twisted up. The outside is merely the space where there is no rubber. Of course, we may choose to focus on the surface as our main interest, but it is unwise to forget that the solid filling is there and to ignore it entirely.

Yes and no. All faces must be contiguous, unbroken surfaces, so holes made by cutting or piercing the surface are forbidden. But some faces can be twisted up to leave toroidal holes in the middle even though the Euler value may disagree. And there can sometimes be internal cavities where the rubber wraps around some region of empty space. See Filling Polytopes.

Duality as we understand it is an unhappy blend of disparate notions. Structurally it is a property of any graph or abstract polytope, however abstract theory allows a broader class of structures than does graph theory. See Polytopes, Duality and Precursors. Geometrically, it is a deep theorem of projective geometry and the associated polar reciprocity. As such, its home is projective space and when we try to trasfer it to Euclidean space it is badly broken and we must develop a Euclidean definition. This last is broadly similar to projective polarity except that the edge, polar to a vertex, is that segment, of some line, which does not cross infinity (and likewise for faces). Commonly, the polyhedron is reciprocated about a concentric sphere or, if there is no centre of symmetry, the centre of gravity (average position) of its vertices; the result may be called the standard dual of that polyhedron and, if all edges of both polyhedra are tangent to the sphere then it is known as the canonical dual. Duality and reciprocation do not affect the interior characteristics of a polyhedron. See Dualising Polyhedra.

Yes and no. Reciprocation is a purely geometrical operation whose home is in morphic projective space. Here, the answer is yes. But in other spaces the principle of reciprocity is broken and it can result in degenerate figures in which elements of the dual polyhedron may be duplicated or missing and so the answer there is no. For example Wenninger insisted on preserving Euclidean symmetries with his "hemi" duals, and this broke their polyhedral structure. Such a degenerate image can sometimes still have a valid "morphic" structure, for example if Wenninger had abandoned his demand to maintain Euclidean symmetries, in which case it is may be morphic polyhedron but not a "faithful" geometric one.

Stellation is the process of extending the faces of a polyhedron until they meet at new vertices to form a new polyhedron or set of polyhedra. The topology of the resulting figure may be radically different from the original. Facetting is the dual process of reducing the interior angles at each vertex to create new faces which form a new polyhedron or set of polyhedra. Both processes require a concrete or metric space such as Euclidean or one of the non-Euclidean geometries, in which flat sub-spaces can be constructed. They are thus unrelated to the abstract and morphic principles which underlie the previous answers. The essentially metric approach of spatial decomposition into cell sets is a symptom of this alien nature and is best discarded as anything other than a practical aid to stellating some polyhedron. Polar reciprocity is a more fruitful approach, with for example stellation diagrams being dual to corresponding facetting diagrams.

This remains my only outstanding question. Morphic theory acknowledges Bridge's finding that several distinct star polyhedra may share the same outward form and that elements may overlap or even coincide. This adds significant complexity to any enumeration. My longstanding approach of working with precursors is confirmed as offering a sound theoretical foundation. One might wish to distinguish between a true *stellation* or facetting which encloses the original core region, from say a *constellation* comprising several discrete polyhedra. We might also define a *cage*, in which individual polyhedra are disposed externally to the original core but are also tangent to each other or intersect. Typically, a constellation will be dual to some cage, but the reverse is not always true. The clash between polar reciprocity and metric expectations cannot be smoothed over, nor can cherry-picking properties from each offer any rigour. To what extent do we wish to abandon intuitive ideas of what I call tidiness? See Tidy Dodecahedra and Icosahedra. It may well be that the answer is whatever you want it to be.

Only the last two questions deal with stellation – the rest are all to do with the general theory of polyhedra. You can perhaps now understand why it is taking me such a long time to get to the bottom of things, especially as I am not in any way a professional mathematician. And I still have to apply my new understanding to the Platonic solids (the regular tetrahedron, cube, octahedron, icosahedron and dodecahedron).

Morphic theory has proved key to bringing everything together into a coherent picture. In essence it extends the topological approach by, as mentioned, allowing faces or cells which are not simple, while embroidering the abstract approach with a more developed theory of geometric realization. Moreover, it does these in such a way as to bring them together to yield a consistent overall picture. Together with the long-established discipline of projective geometry, this has been resolving many more puzzles and inconsistencies than originally foreseen.

My theoretical reappraisal is now largely complete. It 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!

However my investigations into the application of stellation and facetting are progressing at a snail's pace. At least morphic theory provides me with a starting point, which is more than Kepler and his successors ever had. But where polyhedron theory is grounded in the morphic world of topology and projective geometry, stellation theory is grounded in the metric world of Euclidean space. They sit uneasily together (which is surely why over two thousand years of study have still left us floundering). This part of the journey is going to be a very different one.

If you get fed up waiting or prefer to tread the path for yourself, here is some suggested reading, broadly in order of progress from the ground up:

Hilbert, D & Cohn-Vossen, S; *Geometry and the imagination*, 2nd ed, Chelsea (1999)

Coxeter, HSM; *Projective geometry*, 2nd edn, Springer Verlag (1974)

Richeson, D; *Euler's Gem - The polyhedron formula and the birth of topology*, Princeton (2010)

Stewart, B; *Adventures among the toroids*, self-published (1970).

Lakatos, I; *Proofs and Refutations: The logic of Mathematical Discovery*, CUP (1976)

Abrams, L. and Elkind, L.D.C.; "Word Choice in Mathematical Practice: a Case Study in Polyhedra", *Synthese*, Vol. 198, (2019). pp. 3413–3441.

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; "'New' uniform polyhedra", *Discrete Geometry, in Honor of W Kuperberg’s 60th Birthday: Monographs and Textbooks in Pure and Applied Mathematics* **253**, Dekker, New York, (2003), pp. 331–350.

Grünbaum, B; "Graphs of polyhedra; polyhedra as graphs", *Discrete Mathematics* **307**, (2007), 445 – 463

Inchbald, G.; "Morphic polytopes", ongoing web page.

Inchbald, G; "Morphic polytopes and symmetries". In Darvas, György (Ed.); *Complex Symmetries*, Birkhäuser, 2022. Pages 57-70. Hardcover ISBN 978-3-030-88058-3. Softcover ISBN 978-3-030-88061-3. eBook ISBN 978-3-030-88059-0. DOI https://doi.org/10.1007/978-3-030-88059-0.

Gailiunas, P & Sharp, J; "Duality of polyhedra", *Internat. Journ. of Math. Ed. in Science and Technology*, **36**, No. 6 (2005), pp. 617-642.

Grünbaum, B & Shephard, G; "Duality of polyhedra", *Shaping Space – A Polyhedral Approach*, ed. Senechal and Fleck, Birkhäuser (1988), pp. 205-211

Wenninger, M; *Dual Models*, CUP, 1983

Coxeter, HSM et. al.; *The Fifty-Nine Icosahedra"*, 3rd edn, Tarquin, (1999).

Bridge, NJ; "Facetting" the dodecahedron, *Acta Crystallographica* **A30** (1974), pp. 548-552.

Inchbald, G; "Facetting diagrams", *The Mathematical Gazette*, **90**, No. 518, (July 2006), pp.253-261.