Updated 21 Jan 2023

This is the last major topic I need to write up in my theoretical study of polyhedra. The material is all there in my head but jumbled up, like a Bach fugue that has been dropped on the floor, mixed up with a scatter of rejected drafts, and the whole lot thrown in a box together. Rather than wait any longer before putting up my more important conclusions, this is my beginning at getting it all into some sort of order. With luck, it should grow with time.

In getting even to this point, I should like to acknowledge the help and encouragement of many polyhedronists over the years, especially Paul Gailiunas, Branko Grünbaum, Norman Johnson and John Sharp. Sadly, some of them are no longer with us.

Contents

- Notions of duality
- Convex Euclidean polyhedra
- Topological or graph duality
- Projective dualities
- Abstract duality
- Standard and canonical duals
- Morphic duality
- Breaking reciprocity
- Vertex figures
- References

The way in which polyhedra team up as dual pairs, exchanging faces for vertices and vice versa, has been remarked on since antiquity. The cube has six faces and eight vertices, the octahedron eight faces and six vertices, both have twelve edges. Many attempts have been made to state a general theory, but oddities and exceptions always creep in, and none has proved wholly satisfactory. Only a few short studies have ever tried to address such difficulties, and have made only limited headway. It seems to me that over the years, the root of the problem has been that we have approached polyhedra from various different theoretical directions, with the fundamental question as to “what is a polyhedron?” remaining to this day a topic of endless debate and confusion. Each approach brings its own understanding of polyhedral duality, and these dualities are no more consistent with each other than the approaches themselves are.

I would summarise these approaches, broadly in historical order of development, as:

- Convex and highly symmetrical solids
- Graph connectivity and topological surfaces
- Projective geometry, especially projective duality
- Combinatorial or abstract sets

These four approaches – convex, topological, projective and setted – have all proved to be incompatible with one another, and the impact of this on duality is what I shall be exploring here. Some of the root problems only manifest in higher-dimensional “polytopes”. I will try to ensure that my discussion is consistent with polytopes in general, while avoiding such complications as far as possible.

But before going any further, just what are these polyhedra we want to discuss? The term has been much bent or redefined, some may say abused, for the convenience of one theory or another over the years. Many incompatible definitions may be found across the literature, all applicable in their own sphere of interest. [Lakatos 1976]

Here, polygons and polyhedra are understood in their historic geometrical sense, as respectively two- and three-dimensional examples of the more general polytopes in any number of dimensions. A zero-dimensional polytope is a monon (singleton) or point. A one-dimensional polytope is a ditelon (two-ended) or line segment. A four-dimensional polytope is a polycell or polychoron, and so on in higher dimensions still.

Broadly speaking, we can say that a ditelon is a closed segment of a 1-dimensional line. “Closed” in a geometric context means that it includes its two bounding points. In a polyhedron or polytope we call these points vertices and may treat them as monons. A polygon is, or is bounded by, a closed loop or circuit of ditela lying in a 2-dimensional surface. A polyhedron is, or is bounded by, a closed surface made up of polygons and lying in a 3-dimensional space. Crucially, just two ditela meet at each corner or vertex of a polygon, and just two polygons meet at each edge of a polyhedron. Similarly, just two polyhedra meet at each face of a polychoron, and so on. This is sometimes called the dyadic property.

This definition is not rigorous, nor even complete. For example I have not said whether or not polygons must be flat, nor whether the interior enclosed by the boundary is an integral part of the figure. Each theoretical approach brings its own definitions of such things. Several will be encountered later on.

Duality too is a somewhat misunderstood concept that bears some clarification. It is not confined to polytopes but is a widely observed phenomenon. The Yin and Yang of Eastern philosophy, in which every aspect of an idea is reflected in its opposite, offer perhaps the earliest expression of its generality. Mathematical transformations offer many modern examples of dualities, such as the Fourier transform creating a duality between sound waveforms and their frequency spectra, or the ADS/CFT gauge/gravity duality of theoretical physics. So just because you find a duality in one theory of polyhedra, this does not mean that it is the same duality you found in another theory.

Polyhedral duality is slightly different from many such transformations, in that there is no distinct inverse transform; applying the same transform to the dual recovers the original figure. In this it is more like reflection in a mirror or the inversion or reciprocation of a number.

To construct a dual polyhedron we make its vertices into faces, and vice versa, and redraw the edges between them accordingly. But beyond that, the devil is in the detail. It is not always as easy or as clean as it looks.

Duality was first noticed among the highly symmetrical Platonic figures; the cube is dual to the octahedron, the dodecahedron to the icosahedron, and the tetrahedron is self-dual. Euclid developed the theory of geometry which bears his name and constructed these polyhedra in his *Elements*, but for the next two thousand years their duality remained an unexplained mystery.

To Euclid, a polyhedron was a flat-faced solid which existed in the infinite, smooth and contiguous space which we name after him. Only convex examples were studied. That is to say, you could draw a line from any point in the solid to any other, and the whole of the line would lie within it. Every face was a convex polygon.

The most symmetrical forms, principally the regular convex or Platonic solids, have entranced geometers since the earliest times and spurred much of the early research. These symmetries lay at the heart of early observations of polyhedral dualities. They would continue to influence polyhedral thoughts and habits long after their independence from their associated polyhedra was established. Dualities became apparent within these symmetries, most evident in the construction recipes for the regular polyhedra, known as Schläfli symbols; {3 3}, {3 4}, {4 3}, {3 5} and {5 3}.

For a given polyhedron, its dual was typically constructed by taking the centre point of each face, joining these points with new edges, and these edges with new faces, thus creating the dual inside the original. One might consider this an early form of constructing a Euclidean dual.

From the days of Descartes, it has become customary to apply *x*, *y*, *z* coordinates to Euclidean space and to analyse constructions in it accordingly. In this scheme, symmetries are represented by transformation matrices which convert one set of coordinates to another.

In recent times the idea has arisen of dividing space into two halves by the plane in which a face lies. The polyhedron lies inside one of the half-spaces and outside the other. We can then define the polyhedron as the region formed by the intersection of the “inside” half-spaces created by every face. This has become an extremely powerful general mathematical tool, embracing anything from the linear programming of mixing up optimal chemical or agricultural feedstocks to the calculation of particle interaction probabilities in the strange twistor space of high-energy physics.

Dualsing such a polyhedron, defined by half-spaces, is a laborious process which abandons the half-spaces as such and must instead use their bounding planes to derive suitable points and/or lines from which to construct a new set of half-spaces. In practice it may be easier to define the dual by its set of vertex points and then to construct its face planes. This also sits well with the idea of treating space as comprising all the points identifiable by their Cartesian coordinates.

The whole thing is still quite awkward. Only the most symmetrical or simple of polyhedra exhibit dualities which are immediately obvious. If you dualise a polyhedron, then dualise the dual using the same construction method, most such methods will not recover the original figure but one of a different size, perhaps even distorted in some way. There is no elegant general treatment within this mathematical model, which is perhaps why Cartesian coordinates remained the only significant advance for two thousand years. In the last few centuries geometers have found neater shortcuts, but they have had to turn to alternative mathematical treatments to do so.

It was not until 1750 that Leonhard Euler published his pioneering formula relating vertex, edge and face counts, *V*−*E*+*F*=2. He happened to be the first to study two distinct phenomena, the connectivity of planar graphs and the edges of convex polyhedra. His formula turned out to apply equally to both. This discovery signalled the birth of a new theoretical approach to topological structures such as graphs and complexes on manifolds. He called it *Analysis situ* but we know it better today as Topology.

Dualising a graph is easy enough. We mark a new vertex in each region of the graph, not forgetting the region outside it, and join pairs of adjacent vertices across the separating edge. Dualising a polyhedron or a graph turns out to exchange the values of *V* and *F* in the formula, leaving the result unchanged.

But not all polyhedra are convex and these others turned out to yield various alternative results for his formula. This Euler value χ proved to capture certain characteristics of the surface. For example all polyhedra with a hole in have χ=0. This presently gave rise to the discipline of topology, in which the structural form of the figure is of fundamental interest, while matters of size or proportion or position in space become irrelevant. The Euler value and other related properties came to be treated as the defining characteristics of a given topological form or manifold. Ultimately, a polyhedron could be treated as a graph drawn on some associated manifold. Since dualising a figure does not change these defining characteristics, the dual of a polyhedron is then just the dual graph drawn on the same manifold. [Grünbaum 2007]

In this sense any quadrilateral hexahedron, constructed with three faces at each vertex, is topologically equivalent to a cube; we say that they are all isomorphs. They are all topologically dual to any triangular octahedron which is isomorphic to the regular octahedron.

If we wish to establish any kind of dual correspondence between particular geometric figures, topological duality is a prerequisite. Indeed, it has become the defining principle of polyhedral duality in general. But on its own it is not sufficient to define a unique dual for any given geometric figure. For this, properties such as scale, proportion and position must be derived from some other geometrical principle.

A quite distinct approach to duality appeared through the 19th century, with the development of projective geometry.

Edwards (2003) treats the subject in an unusual fashion, being synthetic and practical rather than analytic in tone – indeed you will find more philosophical remarks or construction procedures than equations in his book, which is none the less rigorous for that. I have found this approach to be particularly useful in encouraging the mindset required to follow through the ideas which I present here. The reader who struggles with this section may benefit similarly.

If you apply some projective construction to a point, which transforms it to another point, and find that applying the same construction to the new point transforms it back again, then you have established a *reciprocity* between the two points.

An example is provided via the existence of a *polarity* in the plane about a conic curve. A circle is a convenient example of a conic. Any point has a corresponding polar line, and vice versa. Such a point is called a pole, and the line is its polar with respect to the given conic. Take any point on the polar, and its own polar will meet the original point. This sets up a *polar reciprocity*, in which the two poles and polars form a reciprocal pair.

In three dimensions, a polarity may be establiched about any quadric surface, usually a sphere. In this case the polar is a plane and a reciprocity may again be again established between poles and polars.

As the axiomatic form of projective geometry took shape (equivalent to Euclid's postulates), it established the well-known theorem that you can exchange any statement about points and planes in space with a dual statement about planes and points. A polarity is an example of such a duality.

Thus, polar reciprocation sets up a duality between the reciprocal constructions. It follows that reciprocating any polyhedron about a concentric sphere constructs a projective dual – the dual polyhedron.

Closely associated with the fundamental nature of projective duality is the topology of projective spaces or manifolds. In projective geometry, every line in the plane meets every other line in just one point, while every plane in space meets every other plane in just one line. Unlike the Euclidean, it has no concept of parallelism, no “parallel postulate”. Euclid's idea that parallel lines never meet turns out to create a space almost identical to the projective, except that the place where these lines would have met in projective space must be ripped out of Euclid's. This removes a line from the projective plane, known as the Absolute line, and similarly an Absolute plane from projective space. They leave behind a yawning gap, a discontinuity which can fortunately never be encountered in Euclidean geometry because it lies "at infinity". There is no such gap in a projective space, which therefore wraps back on itself as an unbounded manifold. This gives projective and Euclidean geometries subtly different qualities.

Properties of this Absolute can sometimes be useful in projective geometry. For example the polar of a circle's centre is the Absolute line, while that of a sphere's centre is the Absolute plane. Thus, projective polarities in Euclidean geometry are not universal.

In the 20th century yet a fourth approach emerged. Set theory became fashionable in mathematics and geometers began reformulating space as a set of points. Different kinds of space led to different kinds of set. Euclidean space was a dense point set of infinite size. Finite-set spaces could be created by allowing only a finite number of points. Polyhedra were naturally treated likewise, and to keep them manageable only the vertex points were considered. An edge was treated as a point pair, a polygon as a set of such pairs and the polyhedron itself as a set of such sets of pairs. The combinatorics of these vertex point partially-ordered sets (posets) came under much scrutiny.

But dualising such a set was a nightmare. What had been a vertex became, combinatorially, a cyclic set of cyclic point pairs, i.e. the cycle of faces around it, and vice versa. It was a nasty step backwards.

Eventually work on this, and a more general approach to incidence complexes comprising points, line segments and plane regions, developed into the idea of an abstract polyhedron as a hierarchical set of elements, where each edge or face was treated as an element of the set in its own right.

In the study of incidence complexes, focus is laid on the connectivity of the various elements – vertices, edges, faces and so on – which make up the complex (such as a polyhedral surface). This pattern of connectivity may be represented as a poset of members or elements, in which the elements are ranked by their dimension and connected to elements of one dimension higher or lower by a binary or pairwise "incidence" relation. Certain rules apply, especially to ensure the dyadic property, which is also known as the diamond condition after its characteristic appearance in the set diagram. Such a poset is known as an abstract polyhedron or, more generally, an abstract polytope. A set diagram of any abstract polytope may be constructed, in which the elements of a given rank lie horizontally aligned in a row, the ranks are ordered by dimension, and each incidence is drawn as a line connecting the two elements. This is known as its Hasse diagram.

As I note in *A Critique of Abstract Polytopes*, at first the combinatorialists still treated each abstract element as a set of vertex points. However such an understanding turned out to be an unnecessary complication and the set theory of abstract polytopes is most elegantly expressed if each element is atomic, i.e. a fundamental object in its own right.

In order to construct a real geometric figure, the abstract polyhedron is “realized”, i.e. envisaged as being injected or mapped into some containing real space, usually Euclidean.

Now, here’s the killer. In order to obtain the dual of an abstract polytope, the order of ranking is simply reversed. Informally, one can regard this as turning the Hasse diagram upside down. That is it! In a strong sense, both geometric dual figures share the same abstract structure, and choosing which of the dual pair to realize is just the choice of which direction to adopt for the ranking order.

This kind of duality was originally referred to as combinatorial duality. However the set-of-sets approach is not directly compatible. Moreover the atomic model of abstract posets lends itself to a powerful algebra all its own and, although a relatively recent development, the theory is evolving rapidly.

In *Polytopes, Duality and Precursors* I refer to the undirected abstract set as a *precursor*. This, together with its two dual realizations, forms an overall *schema*. This in turn may be compared to the only slightly longer-established understanding of cuboctahedral symmetry giving rise to the dual pair of cube {4 3} and octahedron {3 4}, with icosidodecahedral symmetry doing the same for {3 5} and {5 3}.

The duality of all geometric polyhedra, by which every polyhedron is twinned with a dual, or reciprocal, geometrical figure, has become a common tenet of mathematical folklore. In particular, the notion that we may reciprocate any polyhedron about a concentric sphere, and thus obtain a suitable dual, is so widespread that it defines what I will call the standard dual of any polyhedron. The standard dual may vary in size, according to the radius of the chosen sphere, but its geometry is otherwise fixed. The Dorman Luke construction is commonly used for uniform polyhedra.[Cundy & Rollett (1961)]

One manifestation in particular has been elevated to canonical status. Any given polyhedron which is a topological sphere (i.e. χ=2) can be morphed into a convex form which has an intersphere, tangent to all its edges, without changing its structure. This tangent solution for each polyhedral structure is is unique and is known as its *canonical form*. The Platonics and Archimedeans are examples of such canonical forms. The polar reciprocal with respect to the intersphere is then its canonical dual, and the whole arrangement forms a canonical dual pair. The Catalans or Archimedean duals are also canonical. (The whole arrangement together with its common abstract poset might be regarded as a canonical schema.)

Extending these ideas to less straightforward polyhedra begins to create problems. Studies exploring some of these issues include Wenninger (1983), Grünbaum & Shephard (1988) and Gailiunas & Sharp (2005).

Wenninger assumes central symmetry as given, and consequently runs into a conflict between projective and Euclidean infinities when dualising the uniform hemipolyhedra, self-intersecting polyhedra which have faces passing through their centre of symmetry. In consequence his dual figures have vertices at infinity, a solution which he recognises as mathematically flawed and not strictly polyhedra any more.

Grünbaum & Shephard observe that much widespread understanding is no more than unproven folklore and often wrong. In particular, once away from the strict assumptions of convexity and symmetry, things can go awry. Had they accepted a wider view, allowing non-convexity and self-intersection, they acknowledge the effects they note would not have been problematic and the insoluble dilemmas would have found solutions. Curiously, while they acknowledge this, they feel the need to assume a convex mindset nonetheless.

Gailiunas & Sharp similarly bemoan the lack of either quality or quantity in the published literature. They consider combinatorial duality in the context of planar graphs (i.e. convex polyhedra), and point out certain difficulties that arise with the early Euclidean process when less simple manifolds (i.e. χ≠2) are considered. They go on, using polar reciprocity, to give examples of solutions which Grünbaum & Shephard did not allow themselves. Their examples amply demonstrate that, in the general case, neither local (non)convexity nor self-(non)intersection is preserved under polar reciprocity.

These short studies all share a common theme, that once convexity and symmetry are left behind, polar reciprocity renders our conventional Euclidean-based understanding hopelessly naive. However none of them offers any systematic understanding of the inconsistencies through which the various difficulties arise.

An example which they do not touch on may be found with the regular star polyhedra. These also form standard dual pairs which may, moreover, be constructed around a common intersphere, however they are not canonical in form; the canonical forms of the great stellated dodecahedron and great icosahedron are respectively just the regular convex dodecahedron and icosahedron, while the small stellated and great dodecahedra are not topological spheres and no canonical form is defined for such polyhedra.

At this point it is worth a quick recap of what this standard approach is actually doing. First of all, its aim is to construct the combinatorial or abstract dual of some original polyhedron, while preserving its overall surface topology. At the same time we wish faces to remain flat, edges straight, and symmetries in Euclidean space conserved. The chosen method is the polar reciprocation of the original about a concentric sphere.

So it is mashing together several different branches of mathematics, each with its own approach to polyhedra and to their duality, in the hope of achieving a consistent picture. Perhaps the most surprising thing, when expressed as I have just done, is how far this endeavour can actually succeed with convex polyhedra. Less surprising on reflection is the number of difficulties which can arise in less straightforward circumstances, where the disparate fundamental assumptions of the various approaches create inconsistencies between then.

In seeking a consistent understanding I have found that both what we take to be a “polygon” or a “polyhedron” and how we choose to “dualise” them have profound consequences. Further consideration of these issues leads on to revisiting our ideas of how to define a polyhedron, its inside and outside, and even of exactly what kind of space we wish to dualise our polyhedra in.

All this was part of my motivation for developing my theory of *Morphic Polytopes*, including morphic polyhedra. A morphic polyhedron may be understood as a graph drawn on some associated topological surface or manifold. It represents an intermediate stage in the realization of an abstract polyhedron as a geometric object, as an *interpretation* of the abstract elements as vertices, edges and faces ranked in a given order. The dual morphic polyhedron is then the same abstract set but interpreted in the reverse ranking order to yield the dual graph on the same manifold.

The second stage of realization is to *concretize* the polyhedron in some geometric space. If we then reciprocate it about a quadric such as a sphere, the reciprocal manifold will have the same topology as the original, while projective duality ensures that the abstract polyhedron is preserved, albeit realized with reverse ranking. Thus, the reciprocal figure is an example of the dual polyhedron.

This procedure does not guarantee that the dual figure will be a *faithful* realization of the abstract polyhedron. Problems are especially likely to arise in Euclidean space.

The abstract formulation poses another difficulty in that it allows more general structures than in topology. Suffice to say here that morphic theory offers a common understanding, in which any inconsistencies are resolved at the interpretative level of realization.

This approach harmonises the topological, projective and abstract approaches to polyhedra and their duality. But it only works flawlessly where polar reciprocity is a consequence of a deep theorem, in other words in projective space.

We usually like to work in Euclidean space. This can be constructed as a projective space with a Cartesian metric applied and the Absolute plane, polar to the coordinate origin, removed. As a direct consequence of this last, the projective duality of statements about points and planes no longer holds as a theorem.

This was the issue encountered by Wenninger with his hemipolyhedron duals. but it is more pervasive than that. Any arrangement which has an undercut, in which the outside of a face is oriented towards the centre of reciprocation, will reciprocate across infinity. This applies for example to asymmetric situations, where the centre of the sphere may even be outside the polyhedron, and also to some star polyhedra which self-intersect in awkward ways.

This is not usually appreciated, as there is an unconscious trick we play which hides the problem from us. Like a sphere, projective space has no boundary and a line in it makes a closed loop around the space. However unlike a sphere, any line not in a given plane must meet that plane exactly once. Thus, when we make the space Euclidean, every line extends across space but is broken where the plane at infinity has been ripped out. Given a finite line segment in Euclidean space, the line it is on extends in both directions; it has two infinite segments as well as the finite one. But if we return it to projective space, these two infinite segments rejoin and the line is divided into only two segments, of equal status.

Given the reciprocation of some arbitrary construction which results in one of these segments, it is equally arbitrary which one it will be. Thus, back in the Euclidean domain, our reciprocal figure might or might not suffer Wenninger's fate.

There are other subtleties of inside vs. outside at work here, which I hope to expand on in due course.

In practice we ignore all this and simply pick the finite segment for our dual polyhedron, along with the face region which the finite segments of the various edge lines bound. The standard dual of an undercut polyhedron is not in fact its projective reciprocal, but a fudge fabricated through our own ignorance. So let us be more accurate and say that the standard dual is that figure which arises from a polar construction, where we choose the segments and faces which do not cross infinity.

We get away with the trick because the choice of segment and region does not affect the connectivity of the elements, and therefore leaves the abstract and topological forms intact. But we break strict projective reciprocity nonetheless.

Wenninger's hemi polyhedra lie on the borderline between star domains and undercuts, and so both segments of the reciprocal extend in opposite directions to a second point at infinity, while neither actually crosses it. But that point at infinity is outside space, and so the polyhedral structure is irretrievably broken. If you move the centroid of the sphere a little so that it no longer lies in the face plane, then the vertex reappears at a finite distance and you can construct a finite dual in the usual way. In a projective space the structure is preserved throughout.

Another problem arises if we wish to preserve the symmetries of the original polyhedron. The polars of its faces and vertices extend equally in all available directions and it becomes entirely arbitrary which we choose for the dual. Wenninger settled for edges in both directions and faces in prisms between them. Since their structure was already broken, this duplication hardly mattered.

However in projective space the two directions do meet again, so to prevent the structure from encircling space as a toroid we are forced to choose one arbitrary direction. This breaks the symmetry. Thus we can see what a profound effect the particular space we are working in may have on our treatment of polyhedra and their duals.

When a polyhedron is dualised, each face is transformed into a vertex figure of the dual, and vice versa. There is thus a duality between faces and vertex figures. This duality is exemplified by the Dorman Luke construction noted earlier, for the duals of the uniform polyhedra.

The principle applies for all polytopes. Every cell is dual to a vertex figure of the dual polytope.

Every face of a polyhedron is represented in i=ts Hasse diagram. The various pieces of the face's boundary - edges and vertices - a are also represented. If we collect together the set just of these elements, they form the Hasse diagram of the associated abstract polygon.

When we dualise a polyhedron by reversing its ranking direction, we also reverse the ranking direction of the abstract polygon buried within it. This subset of the dual polyhedron is realised as the set of elements incident on a given vertex, known as a *vertex star*. But if we ignore the main figure and just number its ranks as a standalone polygon, then we can see that it is a polygon in its own right, the *vertex figure*. We can think of it as a slice through the vertex star.

Abstractly, this vertext figure is still the original abstract face, just with its ranking order redirected and renumbered.

Geometrically, the dual relationship still holds for projective polarisation. The intersection of the original face plane with the polarising sphere is a circle (which may be imaginary). Reciprocating the face in this circle yields a dual vertex figure. The difficulties over crossing infinity and choice of segments and angles do not go away.

- Cundy, H.M. & Rollett, A.P.;
*Mathematical Models*, 2nd Edn, OUP (1961). - Edwards;
*Projective Geometry*, 2nd Edn, Floris (2003). - Gailiunas, P. & Sharp, ; "Duality of Polyhedra",
*Internat. Iourn. of Math. Ed. in Science and Technology*, Vol. 36, No. 6 (2005), pp. 617-642. - Grünbaum, B.; "Graphs of Polyhedra; Polyhedra as Graphs",
*Discrete Mathematics,*, 307 (3–5) (2007), pp. 445–463. - Grünbaum, B. & Shephard, G.; "Duality of Polyhedra", In Senechal and Fleck (Eds),
*Shaping Space – A Polyhedral Approach*, Birkhäuser (1988), pp. 205-211. Reprinted as*Shaping Space: Exploring Polyhedra in Nature, Art, and the Geometrical Imagination*, Springer (2013), pp. 211–216. - Lakatos, Imre;
*Proofs and Refutations*, CUP (1976). - Wenninger, M;
*Dual Models*, CUP (1983).