Posted 30 Aug 2019. Updated 18 Jun 2020, 25 Apr 2021.

Contents

- Origins
- Motivational example
- Manifolds
- Abstraction and realization
- Faithfulness and tidiness
- The morphic synthesis
- Non-simple cells
- Duality
- Regular morphic polytopes
- Compounds, regular or otherwise

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, making little further progress since then. 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. This latest update seems to be shaping up quote well.

But I 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 for certain manifolds in four dimensions) and so despite it being true everywhere else, its applicability must be treated with caution.

Our first mathematical writings on plane polygons and solid polyhedra come from classical Greek authors such as Plato and Archimedes, while hollow bronze dodecahedra were made by the Etruscan ancestors of Rome. So it remained for the next thousand years and more.

When Gottfried Leibnitz conceived of an algebraic analysis of position, his *analysis situs* in the 17th century, he envisaged a new mathematical discipline concerned only with the innate characteristics of structural forms without any reference to quantitative measurements. Leonhard Euler delivered its first stirrings with his famous polyhedron formula *V* − *E* + *F* = 2 and its close correspondence with planar graphs. He called his discipline *stereometry* but while the name did not stick, his ideas did. 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 or a sloping parallelepiped is in essence the same as a cube. Henri Poincaré later developed the field hugely, applying it to general surfaces or manifolds. One of his chief tools remained Euler's decomposition of a topological object or manifold into polygons, polyhedra and so forth, maintaining the foundational link between the two disciplines. Thus, for example, the surface of a polyhedron could be seen as a polygonal decomposition of its associated two-dimensional surface manifold. Poincaré's most innovative and powerful methods also decomposed the entire body of the polyhedron into chains of tetrahedra – three-dimensional simplices – and treating the whole solid as a bounded three-dimensional manifold. In the 20th Century the field 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 Gregor Perelman until the dawn of the 21st century.

From the moment Euler noticed the equivalence with graphs, the focus of polyhedron theory had moved more and more to the surface until the polyhedron became defined as its surface and its body discarded as of no significance. 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 Stewart's topological definition was largely lost lost on his mainstream contemporaries. They were busy developing more fashionable set-theoretic and combinatorial 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 the fans of set theory was Branko Grünbaum, who from the start highlighted the continuous geometric transformations or morphing between examples of a given structural form. This notion of plasticity had always been key to topology (and remains key to morphic theory) however, crucially for Grünbaum, it also provided a rationale for preserving the abstract combinatorial structure of the associated set of points, lines and so forth when individual elements became geometrically superimposed.

Other abstract theorists too were not interested in the exact shape of a figure but only in its "polyhedron-ness" in terms of connected faces and so forth which were incident with each other. In particular Danzer, McMullen and Schulte to some extent paralleled the work of Grünbaum in this field. But they went further, eventually coming up with a rigorous description of such abstract polyhedra in terms of a partially-ordered set (poset) of elements corresponding to the faces, edges and vertices, and abstract polytope more generally in higher dimensions. Moreover the polytope was effectively a set of sets. These subsets comprised cardinal-1 sets each containing a vertex point, cardinal-2 point sets or point pairs (as edges) and cyclically ordered sets of point pairs (as faces), also including the whole set (since every set is a subset of itself) as a maximal element and the empty set (also a subset of every set) as a minimal element. These subsets or elements were partially ordered via a particular pairwise incidence relation between them. The resulting abstract structures belonged to an associated class of more general incidence complexes. Their purely combinatorial connectivity could if desired be "realized" as a geometric polyhedron by injecting it into some ordinary geometric space.

More generally incidence complexes, and especially CW complexes, arose in algebraic topology as less structured assemblages of geometric elements.

Up to solids in three dimensions, modern abstract theory is consistent with the traditional topological approach via decomposition and CW complexes. Beyond that, in four dimensions and above, it turns out to allow more general structures which topologists would throw up their hands in horror at. For example a piece of a topological decomposition must be a simple topological ball, while an equivalent "'*j*-face" of an abstract polytope may be say a toroid or a projective plane. 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. Moreover the process of realization was never pinned down, it has remained a kind of woolly bogey-man in which all the old inconsistencies and disagreements have remained inextricably mixed. Abstract theory has provided a useful algebraic toolset but has introduced more conceptual difficulties than it has resolved.

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 its own right, a bounded manifold. For example a beach ball is a hollow sphere, a 2-manifold, while a bowling ball is a solid, a bounded 3-manifold. 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 ideas morphic theory.

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 typically 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. 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).

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.

Although my original motivation arose from my studies on stellating the icosahedron, the family of polyhedra known as cuploids provide a more convenient example. The pentagonal cuploid shown has sides formed of five intersecting squares and five equilateral triangles. These polyhedra 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 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. However topologically a cuploid is a projective plane, the simplest non-orientable manifold. 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.

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 mean position 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 star pentagon with a hollow central region. 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 the structure at the vertex point itself to be hollow, then it has a hole through it and the polyhedron is then a non-orientable toroid or 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, definitions based on the bounding circuit ignore the whole interior, and abstract theory ignores the entire geometry. Is it a mutilated solid pentagon or what?

In order to resolve such dilemmas, 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, with the boundary 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 base be treated as a topological disc. In the example the pentagon is first to make a star figure, which we may call a Cayley star. It then forms the base of the cuploid. 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 which forms a singularity in the otherwise smooth geometry of the surface. To preserve symmetry this double-wound point is pushed to the centre of the pentagon, shown as *P* in the diagram. In morphic theory, every Cayley polygon has such a point.

Turning now to the vertex figure of the cornoid apex, perhaps the first thing to do is to define it fully in terms of morphic theory. In line with abstract theory it drops the usual topological insistence that the manifold must be disc-like. Noting that topologically a projective plane is a glued-together disc and Möbius band, we choose the Möbius band. (I prefer the less common description as a "band" because it implies a closed loop whereas "strip", as generally used elsewhere, may refer to a flat length with two ends. It is also a band in German, the native language of its discoverers August Möbius and Johann Listing. "Strip" is thus a mistranslation, which it is high time was put right). The illustration below shows how such a band can also be bounded by a polygonal circuit and squashed down into a star polygon. However this time it has 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 may be locally orientable. Since it is double-layered it has density [modulo 2] = 0. Thus, the density value is not expected to match the layer count with Cayley's method equating the two being applicable only to the special case of orientable polyhedra. The cornoid paradox is resolved by the existence of a distinct star pentagon made from a Möbius band. Neither it nor its dual obeys Cayley's density rule, because both are non-orientable.

A morphic *n*-polytope has an interior or body and thus comprises what topologists call an *n*-manifold with boundary, the latter being its bounding surface. The surface itself is a closed unbounded (*n*−1)-manifold. While this surface manifold has often been identified since the 19th century as the polytope itself, this was not originally the case and Poincaré's work on decomposition encompassed the entire bounded manifold or solid, and much analytical topology is still based on this understanding. Topology is thus well able to deal with bounded manifolds, they present no problem in this respect. As mentioned, morphic theory returns to this broader understanding of a polytope as a solid figure.

From this viewpoint the surface, intersections of half-spaces, sets of points and so on are seen as particular aspects which a geometer may choose to focus on and not as general definitions of a polytope.

If the last two 2-manifolds illustrated were zippered or glued together along their boundaries, the resulting manifold without boundary would be a projective plane. This gives some idea of the importance topologists attach to boundaries and to the characteristics of the manifolds they bound.

A polytope can be understood as a graph drawn on the associated topological manifold. The manifold is unbounded and usually finite, although tilings and sponges are sometimes treated as infinite polytopes or apeirotopes. As a graph, it forms what might be called a "proper" topological decomposition of a manifold.

Rigorous topological analysis requires that all the pieces of the decomposition, i.e. elements of the graph, should be simple topological balls. That is to say, the Jordan curve theorem holds and any closed loop drawn on the surface may be shrunk to a point. A disc is a 2-ball, a solid block a 3-ball, and so on. We also include the point as a 0-ball and the line segment as a 1-ball. On a polyhedron, a convex face is a 2-ball. As such the polytope is also an example of what topologists call a chain, and also a CW complex.

However, the hollow star pentagon is a Möbius band and not a disc. The surface decomposition is no longer a chain or CW complex. This is where the cornoid parts company with Poincaré's topological rigour. But it poses no problem for abstract theory.

An abstract polytope is a purely set-theoretic construct which captures the connections or "incidences" of the various elements – vertices, edges, faces, etc. – with each other. The elements are ranked in order of their dimension, with vertices in rank 0, edges in rank 1 and so on. A further partial ordering is provided by their incidences which in the traditional set diagram, known as a Hasse diagram, are shown as joining lines. In this, it is a particular example of the more general incidence complex. In *Polytopes - Abstract and Real*, Johnson describes its properties as monal (each element only occurs once), dyadic (conforming to the diamond condition) and properly connected (no subset is a polytope of equal dimension). The description which follows is somewhat simplified but it will do here; some further clarifications may be found in *A Critique of Abstract Polytopes*.

The illustration shows Hasse diagrams of the four simplest abstract polytope structures (Usually the lettering runs top to bottom, but I have reversed the order here so that each common element retains its label). The null polytope or nullon *A* is just a simple placeholder having no incidence with anything and may be thought of as a kind of equivalent to the empty set. Its rank is one less than 0, or −1. The 0-polytope or monon *B* comprises a point which is incident only with the nullon. The 1-polytope or ditelon comprises a line segment *D* incident with two points or 0-polytopes, *B* and *C*. The simplest 2-polytope or polygon is the digon.

An abstract polyhedron must also satisfy some other conditions, most significantly that a line is incident with only two endpoints, an edge of a polyhedron with only two faces, and so on. This is known as the diamond condition, owing to the characteristic presence of diamond shapes in the Hasse diagram. Such a set is said to be dyadic. The ditelon is the simplest structure able to show this condition, with the monon and nullon being somewhat honorary polytopes. The digon is degenerate in conventional Euclidean geometry but finite in many non-Euclidean situations. Although these four figures are often excluded in working definitions of a polytope, they are important in the procedures used for topological analysis and graph theory as well as in abstract theory.

An abstract polytope is a purely set-theoretic construct. It has no geometric form until mapped or injected into some containing space, a process known as realization. However the form of the resulting geometrical figure need not look much like the abstract structure. The hour-glass-like polyhedron below has four cross-quadrilateral faces and four edges geometrically incident at a false vertex but not structurally incident, which is therefore said to render it "unfaithful." The accurate cube is then a faithful realization.

The awkwardness of introducing unfaithfulness is compounded by valid abstract polytopes which have no unique associated manifold because, unlike graphs or CW complexes, their *j*-faces are not necessarily simple topological *j*-spheres and so the usual topological analysis of the surface can not be applied. What does faithfulness mean for these polytopes? More generally, which mappings of which polytopes preserve faithfulness and which do not? The questions mean nothing while faithfulness remains undefined. Indeed the whole idea of realization has turned out to be just a woolly rag-bag of all the problems, old and new, that had grown up over the years and abstract theory had hoped to resolve.

The abstract theorist's original, vague idea of realization as some kind of mapping turned out to be too broad and undefined for all such realizations to meet their aspirations. Realizations therefore often became judged by an extra criterion of "faithfulness." There is no universal definition of what a faithful realization is either; commonly a geometer might want abstract symmetries to be retained as geometric symmetries and no elements to overlap or superimpose over each other, while others might also demand that the realization of any topological ball be convex or that no elements intersect at all (a bijection). What one geometer finds faithful, another may not, while a given polyhedron may have a faithful realization for one geometer but none for another. For example to some the cuploid is unfaithful because it self-intersects, while say the hemicube is even worse because it cannot be given flat faces without elements coinciding and its symmetry being broken. Is a projective coordinate transformation in a metric projective space (i.e. one which preserves cross-ratios) a reason to reject a metrically-irregular result, when the same transformation in a metric-free projective space leaves it indistinguishable from the original?

From the abstract and morphic points of view, such geometric properties of any concretization are wholly irrelevant. Faithfulness is an afterthought, to be added if desired in whatever form might suit the purpose of some concrete geometer.

Before I came across abstract theory, in my own studies of stellations and facettings I had developed a concept of "tidiness", which is similarly defined according to one's area of interest and which the present context leads me to describe as the geometric quality necessary for the figure to achieve faithfulness to the abstract. That is to say, a tidy geometric polytope is one which is faithful to its abstract structure. The distinction between faithfulness and tidiness is thus one of approach; from the geometrical point of view the figure is tidy, in the abstract view it is faithful. Some further discussion may be found in *Polytopes: Degeneracy and Untidiness*, another work in progress.

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 already 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, of things like size, local curvature or angles. It has only vague things like holes, twists and a decomposition into polygons of equally 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:

**Interpretation**. The various elements of the abstract polyhedron are first interpreted as some kind of geometric entity. They could famously be interpreted as tables, chairs and beer mugs (as would be entirely legitimate), but that would not suit the conventional geometer. The original combinatorial idea of a set of subsets has proved unnecessary and has been abandoned in the most highly abstracted theories, with such subsets being understood as certain "sections" (or subposets or subtopes) of the polytope and quite distinct from the actual elements. Such set-of-sets models may be seen as interpretations of the foundational abstract form. The morphic interpretation also excludes the infinite lines and planes to be found in the theory of configurations, as they are not consistent with the decomposition of a smooth manifold. It interprets the main elements strictly as points, line segments and surface regions, with the maximal element as the interior or body. The abstract incidence relation is deemed to be a geometric incidence or connectivity between adjacent elements.**Concretization**. The containing space, together with any metric it may have, must now be produced and the morphic polyhedron injected into it via a second, concrete mapping. The result is a geometric polytope. Because a morphic polytope is already in some fundamental sense geometric (in the same way as projective geometry), a particular example of such an immersion is called here a concrete polytope. To aid in this the polyhedral manifold (i.e. the interpretation) may also be given an innate metric of its own, in which case the mapping between the two metrics will also need to be defined. Geometric alignment, overlapping or coincidence of distinct elements is in principle allowed, while retaining their individual structural identities.

Morphic theory thus understands or interprets the abstract partial ordering via the incidence relation to be a certain piecewise decomposition of some topological smooth manifold. It is this piecewise manifold which is the morphic polyhedron.

This approach to realization leads us to a taxonomy of polytopes and related constructs along the lines shown in the diagram. The dyadic property is Johnson's term for the diamond condition.

A fundamental difference between the topological and abstract structures now becomes apparent. The abstract construction of the polytope need not be a topologically rigorous one after the manner of Poincaré (topologists might say that it is not equivalent to a chain complex, graph theorists would be similarly sniffy). However the morphic approach demands that topological rigour must still somehow be maintained.

The cuploid and cornoid introduced above serve to illustrate the point. In its recognition of two distinct yet valid star pentagons, the morphic standpoint goes beyond both conventional topology and abstract set theory. Yet, at the same time, it 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.

Where topology demands that any piece of a decomposition be a topological ball, abstract theory makes no such demand and allows us to join such non-balls together without restriction, to form a higher polytope. For example using polyhedral projective planes, such as the cuploids and cornoids, constructs what are known as projective polytopes. But this introduces a grave problem because such a polytope is no longer a proper topological decomposition. Such a decomposition must comprise simple topological balls (a circle is a 2-ball) if topology theory is to be applicable. Even if the cells of such a polytope may conform, when itself used as a cell in a higher polytope it does not 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 but it is no longer a proper topological decomposition of either manifold. Thus, the abstract departure from simple balls means that there is in general no longer a unique manifold associated with the decomposition or graph. Since this association is a fundamental assumption of topological analysis, it opens an equally fundamental divide between the two historically closely-intertwined areas of polytopes and topology.

One might therefore be tempted 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. That might not matter if an easy way were found to classify abstract forms as proper or non-ball decompositions, but it has proved deeply elusive and, to date, a failure of one of the higher aspirations of the abstract theorists. Moreover it leaves anomalies such as the cornoid properties unexplained. It also seems inconsistent, or at least overly restrictive, to allow a toroid as an interior but not as a building-block. One could hardly be laying claim to a general theory. The abstract acceptance of non-ball cells presents a model which is too powerful and elegant to ignore. They arise so naturally within the elegant defining properties of abstract polytopes, and are so elusive in their ability to be readily identified by any abstract properties, that they have become accepted by the abstract community as valid polytopes. And if we accept non-ball cells in higher dimensions then, for consistency, we may not reject non-disc polygons. There remains no reason to deny the Möbius pentagon vertex figure as a polygon in its own right.

However the abstract formulation captures only the combinatorics of an element's boundary and not the topology of its interior. There is thus no way for it to yield the information which it has made necessary.

The only way to provide a complete description is to explicitly define the internal topology of each and every individual cell, there is no getting away from it. From the abstract point of view, this information can only be provided at the realization stage. Morphic theory meets this requirement by adding it in the interpretation step of realization. Once this is done, the associated manifold is now fully defined for topological purposes.

The morphic polytope arises as the intermediate stage created by the breaking of realization into distinct interpretative and concretizing steps. Thus, the individual defining of interiors, as required by morphic theory, lies at the heart of the reconciliation between the topological and abstract approaches.

Grünbaum (2006) identifies each abstract *j*-rank with the *j*-faces of some graph or topologically proper cell complex. The morphic approach relaxes this in order to accommodate non-ball *j*-faces. Instead, the inner topology of each *j*-face must be individually specified. This is the key feature of morphic theory which reconciles the topological and abstract approaches within a common framework. I call such a complex a morphic complex. We name its cells as vertices, edges, faces, ..., body and the abstract incidence relation as one of graph connectivity. The assembled cells form a smooth bounded manifold, a morphic polytope. There remains a bijection between the abstract polytope and its morphic realization. Skeleta with hollow faces and other such oddities are not morphic polytopes, though they remain as alternative interpretations of the abstract form.

Morphic theory might happily stop here, but to fully realize the geometric figure it is still lacking any kind of either metric or containing space; the second step of concretization remains to be considered.

The duality of polyhedra - the exchange of vertices with faces - has been remarked on since ancient times. The duality of graphs followed in its wake. Topologically, the dual of a polytope is the dual graph drawn on the same associated manifold. In abstract theory the duality of polytopes manifests in a particularly elegant way. To obtain the dual of a polytope we need only turn its Hasse diagram upside down so that the direction of the ranking order is reversed (Figure 6). The internal symmetries of the polytope are unaffected.

This ease with which an abstract polytope may be dualised has led me to the idea of a modest further abstraction in which the ranking sequence is defined but not its direction of ranking. The ranking direction then becomes part of the realization process. In this picture, a polyhedron and its dual are two realizations of the same abstract polyhedron. In the context of theoretical physics, such a underlying forms common to a dual pair of entities (.i.e. of equations) have recently been discussed and dubbed schemas. The principle is attractive. But there are technical issues over the role of the empty set in abstract polytopes which make the process less clean than one might have hoped. As with tidiness, before I encountered abstractions or physicists I was developing the notion in the context of stellation and facetting, where I refer to schema as precursors.

Meanwhile, the cuploid and cornoid have one more lesson for us. How can a solid star pentagon dualise to a hollow star vertex figure? The answer lies in the nature of polyhedral duality. Abstractly the polyhedron and its dual are the same polyhedron, just realized with reverse dimensionalities. That is to say, if you number the ranks of its Hasse diagram in the opposite direction, you will obtain the dual polytope, there is no need to rearrange the partial ordering at all. The equivalent manifold decomposition to a polyhedron may be treated as a graph drawn on the associated manifold. The dual decomposition, i.e. the dual morphic polyhedron, is then just the dual graph drawn on the same manifold.

Once a polyhedron is concretized, a geometric dual may be obtained via a projective reciprocity about some quadric surface, which in the standard case is a concentric sphere. This raises some fundamental theoretical issues.

Projective geometry is notable for its principle of duality, in which every theorem is accompanied by the dual theorem. Reciprocity is a consequence of this projective duality. At its most fundamental level, projective geometry has no concept of inside vs. outside, of any spatial ordering of elements with some lying between others. There are only points, lines and planes. Finite edges and faces can only appear when we apply ideas of "between" to our interpretation of the incidence structure.

Unfortunately, dualising a graph with respect to the topological manifold on the one hand and polarising a set of points, lines and planes with respect to some quadric of the containing space on the other, lead to different definitions as to where the bits "between" the vertices are. Projective geometry has a habit of throwing its constructions across what Euclidean geometers tend to call "infinity", making the result for some non-convex polyhedra look very odd.

Elsewhere I develop the idea that polyhedral reciprocity is not the same as projective reciprocity. Specifically, when reciprocating a polyhedron, we identify as its dual edge just the finite segment of the line which is dual to its own. This allows the topology of the manifold to be preserved and is thus consistent with morphic theory. (There is still the odd special case, but that goes beyond the present discussion).

The difficulty is compounded by the morphic observation that the graph of an abstract polytope is not necessarily a proper decomposition of the manifold on which it is drawn. Simply reciprocating the edges and vertices and then intuitively back-filling the faces may not be enough. It may be necessary to reciprocate the whole of the manifold surface in order to preserve its topology. This of course is only possible in the general case if every face of the polytope is individually defined, a key demand of morphic theory.

In the case of the cuploid and cornoid a solid pentagonal base to the cuploid builds a projective plane, with the dual apex closing the cornoid. The cornoid vertex is also a geometric singularity because it is double-wound in the same way as point *P* in the cuploid base. We noted briefly that inserting a Möbius pentagon into a projective plane turns it into a Klein bottle. The dual apex now admits a topological hole, such that the point is not only a geometric singularity in the surface but also one in the surrounding exterior space. Although the two cornoids look similar when constructed via a projective reciprocity, the topology of the singularity at the apex differs between them.

I confess to have broken with tradition and gone backwards. Historically, writers have been motivated by the symmetries of regular polytopes to study them first and then generalise to the irregular as an afterthought. My morphic treatment has gone the other way, perhaps fortunately so for I have not been burdened with the usual preconceptions which arise when one's attention is held by the undeniable power and beauty of symmetries. Remarkably, this has led me not only to the discovery of a whole new class of geometric polytopes, but also to the presence among them of multiple new families of regular ones.

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

A given manifold may be decomposed and twisted up in many ways to produce distinct regular star polytopes. Similarly, any regular skeleton of a star polytope may be fleshed out by many different manifolds to yield many distinct regular stars (i.e. the associated manifold is not unique). For example a regular star hexagon {6/2} can be realized in three dimensions as a circuit of diagonals around a triangular prism. By now, I hope it is not obvious to you how the skew polygon should be filled-in with an interior. Two topologically and hence also morphically distinct examples are illustrated.

A suggestion I end with here is that the morphic approach allows a novel treatment of polyhedral compounds. For example the Schläfli symbol {6/2} is sometimes taken to denote a double-wound triangle (left hand illustrations). This concrete realization preserves the hexagonal abstract structure seen in its Hasse diagram but its geometry does not reflect that. The symbol is therefore more often taken to denote a regular compound of two triangles, the hexagram (upper right). But, while this concretizes the desired geometric symmetry, abstractly it breaks up the hexagonal structure into two triangles, thus changing its abstract symmetry.

If we instead treat the compound figure topologically and hence abstractly as a single 2-manifold with a boundary of two disconnected pieces, we may then regard any regular star polygon as a single abstract figure, as in the Hasse diagram shown (lower right), with the number of boundaries denoted by the highest common factor between the numerator and denominator of its Schläfli symbol. The abstract structure is no longer properly connected and so, although it retains its status as a single structural unit, such a polytope compound is still not a polytope and represents a slightly more general class of incidence complex. However which of the figures shown represents {6/2} must still depend on whether we choose the symbol to reflect its abstract or concrete symmetry.

One might perhaps distinguish between a "polygonal compound" such as two overlapping but independent triangles on the one hand, and a "compound polygon" such as a manifold with two triangular boundaries on the other.

The same principle of a single topological body may be applied to compounds of the regular polyhedra, especially when created as stellations or facettings of some other regular polyhedron. Indeed, there is no need to confine it to the regular examples.

Together the properly-connected and compound polytopes represent a slightly broader class of incidence complex and associated manifold, each having their own symmetries (they remain a sub-class of CW complexes). I suggest referring to this class as polytopal manifolds, or polyfolds.