Updated 1 May 2019

Around the turn of the millennium I began to realise the mess which is polyhedron theory today and set out to try and find something better. What is emerging from this work is a foundational theory of polygons, polyhedra and higher-dimensional polytopes which I call morphic theory. It progressed slowly and only in 2015 was I able to firm it up enough to give it a sensible outline, but it was still only an outline and I have continued to be distracted by other things since and have made little further progress. So rather than wait any longer, I want to set out as much of it as I can and keep adding stuff as I find the time.

But I still have to be careful. I am no more above the odd gross blunder than anybody else, which has obliged me to rework aspects of the theory from time to time. Branko Grünbaum spotted my worst one in a draft paper, and a few years later I was able to repay him by spotting nearly as gross an example in one of his own drafts. Meanwhile one of Poincaré's foundational conjectures (see below), the ability to decompose any given manifold into a definite set of simple pieces, has in the 21st century been proved false in four dimensions (and only in four dimensions) and so despite it being true everywhere else, its applicability must be treated with caution.

When Leonhard Euler discovered his famous polyhedron formula *V* − *E* + *F* = 2 he founded a discipline he called *stereometry*, the analytic study of spatial structures whose surfaces are divided into polygons. Henri Poincaré later developed the field hugely, applying it to general surfaces or manifolds, and called it *analysis situ* – the analysis of position. In the 20th Century it became known as *topology* and, through the algebraic formulations of Emmy Noethe and others, came to play a fundamental role in much of modern mathematics. Much of the century was spent proving Poincaré's huge outpouring of ideas – the famous conjecture which bears his name was not proved by Perelman until the dawn of the 21st century. One of Poincaré's chief tools remained Euler's decomposition of a topological object or manifold into polygons, polyhedra and so forth, maintaining the foundational link between the disciplines. Thus, for example, the surface of a polyhedron could be seen as a polygonal decomposition of its associated manifold. Two visually distinct polyhedra with similar connectivity of faces, etc. came to be regarded essentially as different examples of a common underlying form, for example a rectangular box has the same topology as a cube.

In the latter half of the 20th Century Bonnie Stewart studied toroidal polyhedra, for the first time actually defining his polyhedra 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 of philosophical logic and all rational thought. It has since failed to live up to expectations).

Among them was Branko Grünbaum, who initially highlighted the continuous transformations or morphing between such examples of a given topology, treating for example a rectangular box and a cube as distinct morphs of the same combinatorially connected object, in effect a graph drawn on the associated surface manifold. This notion of plasticity had always been key to the discipline of topology (and remains key to morphic theory) but was less obviously related to set theory.

The set theorists were not interested in the exact shape of a figure but only its "polyhedron-ness" in terms of connected faces and so forth which were incident with each other. In particular Danzer, McMullen McMullen and Schulte built on the pioneering work of Grünbaum in this field, eventually coming up with a description of the connectivity in terms of a partially-ordered set (poset) of elements corresponding to the faces, edges and vertices, which they called an abstract polyhedron, or, more generally in higher dimensions, an abstract polytope. This purely combinatorial connectivity could if desired be "realized" as a geometric polyhedron by injecting it into some ordinary geometric space. But the process of realization was never pinned down, it has remained a kind of woolly bogey-man for any set theorist trying to fully define a given geometric polyhedron in abstract terms.

Up to solids in three dimensions, the set-based abstract theory is consistent with the traditional topological approach via decomposition. Beyond that, in four dimensions and above, it allows more general decompositions which topologists would throw up their hands in horror at. In this, certain thorny examples of apparently paradoxical polyhedra have led me to follow the set theorists towards greater generalization. But more of that later.

Essentially I have sought a more satisfactory general theory by following Stewart's approach another step back towards Poincaré's, considering not only the surface but also the interior as a topological object, in this case a bounded manifold, in its own right. Topology is sometimes aptly described as "rubber-sheet geometry", ignoring lengths and angles (except as aids to analysis) and just considering the underlying structural form or morphology. That is pretty much how I have approached polyhedra and so I call my theory morphic geometry.

From the morphic standpoint, the various properties of a polyhedron arise from several distinct disciplines or layers of understanding:

- Abstract properties are concerned with the elements of the polyhedron and the combinatorics of their incidences defining its overall structure.
- Topological properties are concerned with the overall or global nature of the surface manifold onto which that structure is mapped. Morphic theory is most relevant where the surface is a smooth manifold and its interior or body is a similar manifold of one dimension higher, save that it is bounded by that single contiguous surface manifold.
- Morphic properties are concerned also with the way in which the topological manifold is mapped into its containing space, yet another manifold which must also be locally smooth and contiguous in order to render the mapping non-trivial, though the space may not necessarily be locally Euclidean since no metric need be considered.
- A manifold may have its own internal metric. This applies to both topological polyhedra and the spaces they inhabit. A polyhedron's internal metric may differ from that of the surrounding space, in which case some form of transformation may be used to move from the one to the other. Morphic theory is not concerned with this aspect of manifolds. Where abstract theory discards all of the spatial geometry, metric included, topology and morphic theory discard only the metric geometry (save as an analytical convenience).

From the abstract point of view, morphic theory is very much a theory of realization. At least in origin, topology deals in rubber sheets which exist in ordinary space but have no definite size or shape. In this picture, the abstract poset has already been partially realized. But there is as yet no notion of a metric, of measurements; things like size, local curvature or angles, only vague things like holes, twists and a decomposition into polygons of undefined size and shape. This half-realized, rubber ghost of a solid, with its surface decomposed into vaguely-shaped but precisely connected polygons, is an example of what I call a morphic polyhedron or, generalized to any number of dimensions, a morphic polytope. Thus, realization is seen to be a two-step process. In fact, we will later see that there are three distinct steps.

A rubber sheet has a precisely connected interior. But as already mentioned, abstract decompositions go beyond rubber sheets to shapes whose interiors are not even defined. Morphic theory bridges this gap by – but I will introduce that through a motivational example.

For, remarkably, all this has led me to the discovery of a whole new class of geometric polytopes, including multiple families of regular ones.

The family of polyhedra known as cuploids have a curiously paradoxical property. They have the superficial appearance of toroids, except that one side of the central hole is closed off by a membrane. Or is it? The pentagonal cuploid shown has sides formed of five intersecting squares and five equilateral triangles. Topologically a cuploid is a projective plane, the simplest non-orientable manifold. The central membrane, typically the central region of a star polygon, is wound twice around its central point, making it appear to be two layers thick. Arthur Cayley pointed out how we can treat such layers of a self-intersecting polyhedron as surrounding a dense interior, so passing through two layers should give a density change across them of two. Except, in the case of a cuploid, both sides are outside the polyhedron so the density on both sides is zero and the net change across the layers, the difference between them, is therefore 0 − 0 = 0. One might suggest that the two layers are oriented oppositely, giving the true change across the membrane as 1 − 1 = 0, except that the surface is non-orientable so this condition cannot be defined. A second paradox arises when we ask whether this zero-change membrane is therefore a membrane at all. If there is no change in density across it, then surely it cannot exist and there must be a hole there. But a hole would open up a through-hole from one side of the polyhedron to the other, changing it to a different manifold. And the star polygon would no longer be a topological disc, one of the key requirements for (what I will call in this context) a proper decomposition, meaning that the figure was not even a polyhedron any more. Whatever way we try to resolve the paradox, we run into trouble.

Dualising the cuploid rubs in the paradox even more. Its dual is typically obtained by reciprocating about a concentric sphere (for our purpose we may take the cuploid's centre as the centre of gravity of its vertices). This yields a hollow pyramid which I call a keratinoid or cornoid, because it is reminiscent of an old-style drinking horn in the same way that a cuploid is reminiscent of a drinking cup.

The vertex figure of its apex, loosely speaking the section revealed when the vertex is cut off, is without question a hollow pentagon. Like the cuploid, the cornoid is topologically a non-orientable projective plane. This requires that the surface be connected at its apex because otherwise, if we take its vertex to be hollow, then it has a hole through it and is instead a Klein bottle. How, then, can we have a connected vertex whose vertex figure is hollow? How do we even define a hollow star pentagon, as traditional topology ignores the hole while abstract theory ignores the entire geometry. Is it a mutilated solid pentagon or what?

Morphic theory offers a different approach. One may consider any polygon, which forms a face or "2-cell" of a polyhedron, as a bounded manifold in its own right, although only the boundary is decomposed into a polygonal circuit of sides and vertices.

The nature of the cuploid surface, as a topological decomposition of the projective plane, requires that the pentagon be treated as a topological disc. We first twist up an ordinary disc-like pentagon to make a star figure and then form the base of the cuploid with it. There is definitely a double-layered membrane present, and it has a feature which is seldom noticed. As a model of the projective plane, the cuploid is said to be immersed in ordinary space. When the pentagon was twisted round and flattened, it formed a double-circuit - and angle of 4π or 720° - around a point in the central region. To preserve symmetry, this is the centre of the pentagon, shown as *P* in the diagram.

Turning now to the vertex figure for the cornoid apex, perhaps the first thing to do is to define it fully in terms of polyhedron theory. Morphic theory does not insist that the polygonal manifold we started with must be disc-like. It can be whatever we like, for example a Möbius strip or band. The illustration shows how such a band can be bounded by a polygonal circuit and squashed down into a star polygon, moreover a star polygon with a hole through the middle. This matches the polygon seen in the vertex figure and serves as its definition.

The resolution of the original paradox is now straight forward. Because the cuploid surface is non-orientable, the star pentagon at its base inherits this property even though it is in itself orientable. Non-orientable densities do not add in the same way as orientable ones, but can achieve no more than a modulo 2 count, according to whether the number of layers is odd or even, giving: 1 + 1 [modulo 2] = 0. Thus, the density does not match the layer count. The cornoid paradox is resolved by the existence of a distinct star pentagon, made from a Möbius band. Neither polyhedron obeys Cayley's density rule, because both are non-orientable. Density can best be understood as the sum of the orientations of the multiple layers and not as a layer count in itself, with Cayley's method being applicable only to the special case of orientable polyhedra.

In its recognition of two distinct yet valid star pentagons, morphic theory goes beyond both conventional topology and abstract set theory. Yet, at the same time, morphic theory resolves the differences between these two approaches in a way which creates a new and self-consistent model.

Many non-convex polyhedra have interiors which are not topological balls. This is what led to the first extension of Euler's formula, to include toroidal holes or handles and non-orientable twists. The cuploid is one example, Stewart's toroids provide another. Abstract theory allows us to join such non-balls together without restriction, to form a higher polytope.

But such a polytope is no longer a proper topological decomposition of a manifold and, even though its cells conform, it does not itself obey the higher-dimensional analogues and extensions of Euler's polyhedron formula. Its topology cannot be expressed analytically and unambiguously from its abstract properties alone. For example if we substitute a hollow Möbius star for the base of the pentagonal cuploid, we obtain a surface which is topologically a Klein bottle instead of a projective plane, although its sum of V − E + F does not change. It is a valid abstract polyhedron (and a valid morphic one) but it is no longer a proper topological decomposition of either manifold.

It might therefore be tempting to restrict one's attention to proper decompositions and cast one's definitions accordingly, as Stewart did (and indeed, I did until Branko pulled me up). But in that case, one would be rejecting many otherwise valid abstract polytopes and leaving anomalies such as the cornoid properties unexplained. And why allow a toriod as an interior but not a building-block? One could hardly be laying claim to a general theory. Given the consequences of the abstract approach, with its acceptance of non-ball cells, there is no reason to deny the Möbius pentagon vertex figure as a polygon in its own right.

Given this morphic understanding, we can now see that the geometric realization of some abstract figure comprises a three-step process:

- The various elements of the abstract polyhedron are first mapped to some kind of geometric entity. They could be mapped to say tables, chairs and beer mugs (as would be entirely legitimate), but that would not suit the conventional geometer. So they are mapped to points, line segments and plane regions. The maximal element is mapped to a region of space and the minimal element, the empty set, may be either discarded or mapped to the rest of space that is "null" in terms of the polyhedron (hence it is sometimes called the null polytope or nullitope). The abstract incidence relation is declared to be a geometric incidence or connectivity between adjacent elements. You might think that this now defines a topological manifold, for example if the Euler characteristic is equal to 2 then it should define a sphere. But this is not so, for nobody has yet said that the plane region comprising a given face must be simple.
- So the topology of each individual face must now be defined. Usually it will be a topological disc, a 2-sphere, but we have already seen examples where it must be a Möbius band. We already know that the interiors of toroids and non-orientable polyhedra are not simple, so we also need to similarly define the topology of interior. The associated bounded manifold is now fully defined for topological purposes, i.e. the morphic polyhedron is fully defined. but what is lacking is any kind of metric or containing space.
- The containing space, together with its own metric, must now be produced and the polyhedron mapped into it via a second, concrete mapping. If the polyhedral manifold contains an innate metric of its own, the mapping will need to be more rigorously defined. For example are any combinatorial symmetries preserved in the metric of the manifold, and if so then are they furher preserved in the metric of the containing space? With all this sorted out, at last we have our geometric polyhedron.

The sharp-eyed among you will by now have recognised that the Möbius star pentagon discussed here is not only regular but represents an entirely new class of regular star polygons and, by extension, of regular star polytopes in general.