Polytopes, Duality and Precursors

Updated 8 July 2020

The idea of polytope schemas had its roots many years ago in the duality of stellation and facetting. It has evolved over time and this study has been substantially rewritten and renamed – twice. Precursors were once variously called 'generators' or 'templates' but for technical reasosn those terms proved unsuitable. The developing mathematical foundation turned out to be more or less inherent in modern abstract theory and, in recasting it in abstract terms, I discovered flaws in both approaches. More recently the idea has arisen in theoretical physics of a 'schema' with an underlying 'common core' and it seems a useful concept to adopt here, despite the potential of confusion with Schläfli's "polyschema". The theory of precursors, the "common core" of a polytope schema, remains a work unfinished The later parts of this essay which have yet to be updated are highlighted in pink.

Contents

Introduction

Despite over two millennia of academic study, mathematicians have struggled to come up with a satisfactory formal definition of polyhedra and higher polytopes. Grünbaum referred to this failure as "The original sin in the theory of polyhedra" [1]. Nowadays we have come to understand a polytope to be a geometrical "realization", an injection into some containing space, of a particular kind of set-theoretic structure known as an abstract polytope. The abstract construction actually tells us very little about the form of some real geometrical object. Without a rigorous definition of the realization process, we may have little idea what geometrical properties the resulting polytope might have, and this is not much of an advance. And we do not; we still lack a model which accurately captures the properties of the real thing.

The duality of polyhedra and polytopes, and their mutual construction via a polar reciprocity, is well known. Projective geometry and the abstract formulation present elegant models as far as they go, but each is limited in its own way.

This essay explores an alternative model for polyhedra and polytopes, based on the need to define the structure and geometry of vertices as well as faces. Development of this idea leads to a less limited model of polyhedral duality, via a level of order underlying all dualities. In the context of polytopes I refer here to this level as precursors.

But before getting started, I should note some ideas and terms I will be using; others used later on will be introduced where appropriate.

A mathematical space can have many dimensions, in general p. A polytope, or p-tope, is a certain kind of closed geometric figure in p-space. A polytope in zero dimensions is a point or vertex, in one dimension a ditelon, in two a polygon, three a polyhedron, four a polychoron, and so on. A polytope is said to be built up by assembling polytopes of one dimension lower, here called cells; a cell of a polygon is a side, that of a polyhedron is a face, and so on. The examples in this essay seldom venture beyond three dimensions, and it is not necessary to understand those that do in order to follow the main argument.

It is normal to speak universally of vertices and edges. It will sometimes be convenient here to distinguish between the vertices and edges of a polyhedron or higher polytope, the corners and sides of a polygon, and the ends of a line segment.

The topological structure of a polytope is distinguished from its geometrical form. Many problems with earlier definitions arose from the failure to distinguish topological properties from geometrical ones. For example, faces would typically be defined in the same sentence as having both topological properties such as connectedness and geometric properties such as flatness, making it difficult to see which properties might be the more fundamental, or which might be adapted to other situations, such as say bubbles or foams, without damaging the underlying theory. We say that a geometrical polytope is some realization of the corresponding topological polytope. Likewise, the topological object is some abstraction of the geometric one.

The importance of vertex figures

A polytope is sometimes said to be a collection of its surface features, such as points, lines, planes and so on. One problem with this simple view is that the spatial relationships of the various features are not necessarily determined. For example some collections of features can be assembled in different orderings, or topologies, as with the rhombicuboctahedron and Johnson solid J37 (the elongated square gyrobicupola). Some can be assembled in different geometric arrangements having the same topology, as with the convex and great regular dodecahedra, or with the regular icosahedron and the irregular one formed by inverting a "pyramid" of five adjacent faces. In order to distinguish between such isomers we need information about both the relative orientations of the faces, and the way in which faces and edges are connected. For example the two icoshaedra mentioned differ only in the orientation of their faces, while the regular and great dodecahedra differ only in the order in which their faces are connected. All that assumes that say all edges are the same length, which is not usually the case. For example a figure which is structurally a cube may turn out in fact to be a cuboid, a parallelpiped or seomething even less regular.

One way to provide the missing information is via coordinates. However any polyhedron thus defined has a specific scale and orientation in space, which makes comparisons between different polyhedra tiresome and consequently makes generalisation unnecessarily cumbersome. Another way is via angles. But here, once we move into three or more dimensions, solid angles do not have an exact geometry. For example if we assemble four polygons around a vertex, we can obtain a given soild angle in either of two ways, by pushing one or other pair of opposing edges together an appropriate distance, even though each polygon remains rigid. The dihedral angle between faces at each edge will give us the information we need, provided we know which faces are connected along which edge.

A more elegant way to provide all this information is via vertex figures. A vertex figure of a polyhedron may loosely be thought of as the polygonal surface revealed when a corner of the polyhedron is sliced off; many specific definitions are discussed in my essay on vertex figures. The kind of interest here will be defined later. The connectivity of faces and edges around the vertex is shown by the connectivity of the sides and corners of the vertex figure. The dihedral angles between faces can be deduced from the corner angles of the vertex figure and its geometrical relationship with the associated vertex.

The idea of vertex figures generalises to other dimensionalities. In four dimensions the cut surface is a polyhedron, and generally in p dimensions it is some (p-1)-tope. In two dimensions it is just a line or ditelon, which I will sometimes call the corner figure. Different corner figures are characterised by their various lengths. In one dimension the vertex figure is just a point, which I will sometimes call the end figure.

When a polytope is constructed, the geometry of each cell alone is not enough to define the final figure. For example there is nothing to stop one or two caps of five triangles on a regular icosahedron from being inverted into dimples. There are even breathing polyhedra which can smoothly accommodate a range of angles just as a quadrilateral can. In order to fully define the geometry of the whole figure, the geometries of the vertex figures are also needed.

Conversely, the local geometry at each vertex is insufficient. For example they are identical for every cuboid, so these polyhedra cannot be distinguished. It is therefore necessary to include the geometries of the faces.

Thus, to fully define the geometry of a given polyhedron, it is necessary to define that of each cell and vertex figure. Only in this way can the coordinates of each part of the figure be uniquely determined and the geometry of the polytope be fully defined.

Coxeter finds it convenient to define a regular polytope as one having regular faces and regular vertex figures.[2] It is a short step to observe that, more generally, any polytope in general would similarly be defined by its cells and vertex figures. Coxeter chose this definition for its simplicity. It is the great beauty of mathematics that simplicity and power often go together.

We now modify our hierarchical definiton of a polytope accordingly. A 3D polyhedron is now made up of 2D faces and vertex figures. These comprise 1D sides and corner figures, and these in turn 0D vertices and end figures. The hierarchy may be extended upwards to p dimensions.

Abstract polytopes

First, I should warn you that abstract theory has taken a number of standard terms from traditional polytope theory and given them specialist meanings wholly incompatible with their usual ones. I warn where this happens and give as simple account as I can.

An abstract polytope is a set of elements or faces which are partially ordered by a ranking function and an incidence relation between elements of adjacent rank. A polytope of dimension p has ranks j for −1 ≤ jp, with elements of rank j called j-faces. Thus it has two more ranks than its dimension. Note that a "face" can thus be of any dimension, not just two. There is just one face each of ranks p and −1, respectively named the maximal and minimal faces.[7] The standard diagram depicting the ranked incidence structure is known as a Hasse diagram.

Hasse diagram of the abstract cube.

The set-theoretic definition imposes certain structural properties which Johnson[6] has summarised as monal, dyadic and properly connected; these essentially bar monsters such as compounds, objects joined at a single vertex, configurations, and so on.

An abstract face is not a polytope in its own right. Rather, it corresponds to the interior or body of such a polytope; for example a 2-face corresponds to the plane region bounded by the edges of a polygon.

A key distinction is therefore made between a single element or face and the wider subset which also includes the faces of lower dimension which form its boundary. These bounding faces, together with the bounded face and the null face, do form a polytope in its own right. For example a poygonal facet comprises not only a 2-face but also the 1-faces incident with it, the 0-faces incident with those and the −1 face which is incident with them in turn. This gives it the correct ranked structure to comprise a 2-polytope. The null face has no subsidiary faces and is often treated as the (self-contained) null polytope. These sub-polytopes have had various names but nowadays are typically called facets. This usage of the term is inconsistent with the theory of stellation and facetting and one must be careful – especially on this website – not to confuse the two meanings.

This dictinction between faces and facets was not originally made. Abstract theory has its roots in the combinatorial approach to geometric elements as sets of vertex points; an edge was described by its two endpoints, a polygon by a cyclically ordered set of such point pairs, and so on. This led to an abstract theory in which the polytope was a set of such vertex point sets. In set theory a distinction is made between a member or element of a set and a subset of a set. An element is not necessarily a subset (though it might be), while a subset is not necessarily an element (though it might be, as in the combinatorial model). Every set includes both itself and the empty set as subsets, so when it became necessary to include maximal and minimal elements it was a natural step to identfy the maximal element with the whole point set and the minimal element with the empty set.

But the whole subset-based approach lacked rigour because it turned out to be a confusion of faces with their corresponding facets. For example every face included itself as part of the subset, which set up a recursive inclusion of every element of the subset. It also measnt that when isolating a given face, one found onself bringing with it the whole subset. So it had to be modified to recognise the distinction. Many modern treatments of abstract polytopes still show only grudging acknowledgement of the problem.

Realization

The above abstract formulation says nothing about the geometric appearance of the polytope. A geometric figure must be derived from the abstract formulation through some process of realization, by which it is mapped or injected into some containing space such as Euclidean space. Standard abstract theory is somewhat woolly about this process, typically only making demands on the outcome.

Morphic theory recognizes two distinct steps or stages to the realization process:

The first step of interpretation defines the objects that each rank of abstract elements represents. They could be tables, chairs and beer mugs for all that abstract theory has to say about them. But we choose such geometrical entities as points (vertices), line segments (edges), surface regions (faces) and so on. The maximal element is the body or interior of the figure and the minimal element, required by set theory for completeness as the emty set, is left as a kind of null polytope or nullitope. The incidence relation is defined as a structural, and hence also physical, adjacency or connectivity.

The assemblage of all these elements now comprises a topological manifold, a "rubber-sheet" figure. Morphic theory treats it as a bounded manifold, the boundary being the polyhedral surface and the rest of the manifold its body. This contrasts with the more traditional twentieth-century focus on the bounding surface, harking back specifically to the original idea of a solid body.

The second step of concretization is to inject the manifold into some geometric space, typically one with a metric so that we may give it lengths and angles. It is at this stage that we decide whether the faces should be flat and edges straight, as a conventional polyhedron, or all lie on the surface of a sphere as a spherical polyhedron, or everything lies in a plane as a perspective drawing and therefore geometrically degenerate, and so on.

Duality and schemas

With the above conceptual and theoretical background established, we can now begin to approach the core thesis of this essay.

Projective duality and reciprocity

The dualities between points, lines, planes and so on are a theorem buried deep within the foundations of projective geometry. One manifestation is the well-known reciprocity of two dual polyhedra about a concentric sphere.

The exact mathematical correspondence between projective and polyhedral reciprocities breaks down for some non-convex figures and requires subtle modification to restore, but that need not concern us here.

In general, for a polytope of dimension p (where p > 0, since the (−1)-sphere is undefined), reciprocation about some (p−1)-sphere twins it with a dual polytope.

Schemas in theoretical physics

Symmetries and dualities also provide some of the most fundamental underpinnings of modern theoretical physics. Duality itself has come to be seen as a kind of "giant symmetry". One such duality is found in the relationship between gauge field theories of physics, exemplified by quantum field theories, and the less conventional unified gravitational string theories in ten or eleven dimensions.

De Haro and Butterfield have recently highlighted the treatment of duality as a symmetry and from that notion have been developing the idea of an overall "schema" in which a common core, a "bare theory", is expressed in two dual physical theories, gauge and gravity respectively, which are mathematical isomorphs or "isomorphic models" of the bare theory.[5]

They note that their schema principle is extremely broad and has much wider scope than the dualities they happen to be interested in. It is a short step to exploring its relevance to the duality of polytopes, with some kind of common core forming a precursor to two dual polytopes.

Abstract duality

The dual of an abstract polytope is obtained in a remarkably simple and beautiful way, by reversing the order of ranking. That is all. Ranks p and −1 exchange places, as do ranks p−1 and 0, and so on. The incidence relations remain unaffected.

Thus, the same abstract structure underlies both a polytope and its dual, making it an obvious candidate for a polytope precursor. But the current formulation of abstract theory raises a subtle problem.

The minimal element is customarily identified with the empty set. This creates an anomaly when we try to dualise a polytope by reversing its ranking. The empty set now becomes the maximal element. The defining properties of an abstract polytope do not allow an element to occur twice, so it becomes impossible for the minimal element also to be the empty set. The dual is no longer regarded as a valid abstract polytope.

It seems to me that the most elegant solution is to discard the convention that the minimal element and the empty set should be identified. The convention arose with the combinatorial set-of-sets approach but somehow survived the reappraisal, perhaps because it had no corresponding facet to distinguish it from. Another awkwardness with the combinatorial approach had come when dualising. The various subsets could not simply be re-ranked, as the cells had now become vertices and vice versa. The old set had to be discarded and the new vertices relabelled and re-allocated to the various subsets. It was horrible. With the recognition that polytopes are not fundamentally sets of sets, and with the residual complications it causes the dualising process, it appears high time the last residual subset was evicted from the abstract conception.

Directing the ranking order may now be wholly removed from the abstract formalism, instead becoming a part of the first interpretative stage of realization, along with the appearance of geometric objects such as points and line segments.

The resulting unoriented abstract structure, given a ranking sequence but not a direction for that sequence, has both the necessary rigour and the necessary generality to form the basis for a precursor.

Vertex figures revisited

Vertex figures and duality

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 for the duals of the uniform polyhedra (see Figure).

Dorman Luke construction for the rhombic dodecahedron, showing the duality of face (blue) and vertex figure (red).

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

Vertex figures and abstraction

Recall that in abstract terminology, a face element and a facet sub-polytope can be of any dimension. A (p−1)-facet corresponds to a cell and for technical reasons I shall call it a cell section from now on. (It does not matter what an abstract section is in general, except to note that it is not at all the same thing as the well-known sectional view of an object found in the descriptive geometry commonly used by mechanical engineers.)

Dualising a cell section – by reversing the ranking with respect to the whole polytope – yields a dual sub-polytope known as a vertex section. When the polytope is realized as a whole, the vertex section is realized as a vertex star. This comrises the set of elements which are incident on the given vertex, up to and including the body; its ranks within the parent have dimensions 0 to p.

A vertex section has the same number of ranks as a cell section, but it is shifted up one rank overall. If it is realized with the same ranking level as a cell section, i.e. one level lower than the star, then it is realized as the vertex figure. The vertex figure may now be understood as a lower-dimensional slice through the vertex star.

Vertex section (blue), vertex star and vertex figure (red) of the cube.

When a vertex section is realized, its appearance thus depends on the chosen mapping; map it to its full dimensionality and it appears as a vertex star, but map it to one dimension lower and it appears as a vertex figure.

Thus, Coxeter's choice of both cells and vertex figures as definitive does not quite match abstract theory. It would be more direct to define a geometric polytope in terms of its cells and vertex stars. However the vertex star carries a great deal of redundant information, for example many or even all the other elements may be deduced from the shape of the body alone. The vertex figure eliminates all this redundant information.

For this reason I will continue to work with vertex figures. It will be interesting to see whether this ultimately raises any awkward issue.

Polytope schemas

Precursors and schemas

Cell sections and vertex sections are examples of more general sections of an abstract polytope. Specifically, they are collectively its (p−1)-dimensional sections.

The set of p−1 dimensional sections of an abstract polytope thus comprises two subsets, its cell sections and its vertex sections. Which subset is which depends only on the direction or orientation in which its ranks are ordered. This orientation is a part of the first, interpretative step of realization. I shall call the as-yet undirected abstract set of sections a precursor.

The sequence of ranking may be interpreted in either of two directions. These two interpretations yield dual polytopes. The underlying precursor, together with the ordering process and the resulting duals, comprise the schema for the polytope. (At least, this is my understanding of De Haro and Butterfield's model: the abstract precursor is equivalent to their "core theory".[5])

Abstract precursors

Certain sections relating to a face have other names. For some face F of P, the facet F/F−1 is also referred to as the span of F, and the facet Fp/F as the co-span of F. The span of a (p−1)-face is a cell section. The co-span of a vertex is a vertex section. The section F/F is just F. The span of Fp is P and its co-span is Fp, while dually the span of F−1 is F−1 and its co-span is P.

We can now build up a polytope by assembling its elements dimension by dimension. First we assemble ditela from vertices, then polygons from ditela, polyhedra from polygons and so on until we assemble the cell sections to form the completed figure. Abstractly, at each stage we are assembling the spans of the various elements together to create the span of the next element higher.

Dually, we can assemble a description by working down from the top. First we identify the cells, then the cell boundaries, then on down until we reach the edges and vertices. Abstractly, at each stage we are working down assembling the co-spans of the various elements together to complete the co-span of the next element lower.

The spans of one polytope are dual to the co-spans of the dual polytope. They represent a recursive generalization of the duality of cell and vertex sections, with each dimension smaller of spans and co-spans being the cell and vertex sections of the previous, greater level of cell and vertex sections.

This set of spans and co-spans may be shorn of any absolute dimensionality in the same way that the abstract polytope was to create the unoriented precursor. Indeed, it may be regarded as the anatomy of the precursor and an integral part of the schema.

What follows still needs to be revisited and rewritten

Construction of a geometric polytope

It now becomes natural to define a given geometric figure as a realization of its precursor. The precursor must follow the two-step realization process, first being interpreted as one of two dual morphic manifolds and then being concretized with a specific geometry.

The detailed geometry may be defined by following the same assembly process of its anatomy

Polytope precursors

We now have sufficient understanding to attempt a more formal understanding of precursors.

p = −1, the null polytope, is a degenerate case in which F−1/F−1 = F−1. It is thus self-dual. Its realization is undefined but is sometimes treated as that part of space outside the polytope. [So I need to figure how, as an element, its co-span expands to embrace the whole polytope: or, should I just treat this list that way in the first place?]

p = 0, a single point, is another degenerate case. The point precursor is defined as the set {a}, which may be manifest [manifested?] as either of the 0-topes A or A', where A=A'. We can say that any point A is self-dual in 0-space.

The linear precursor, i.e. for p = 1, comprises the set of point schemas {{a} {b}}, abbreviated to {a b}. Its manifestations are the dual lines AB and A'B' [we could perhaps define it as the two recursively connected sets {{a b} {a b}} - is there any advantage?]. Notice that in 1-space the point A is no longer self-dual, but is dual to A'.

For p = 2 or more, a precursor in p dimensions comprises two sets of precursors in p-1 dimensions, which are irreducible and closed under recursion. It may be manifest[ed?] as either of two dual p-topes. [expand on p=2 and 3?].

It is worth a reminder at this point that the polytopes discussed so far are topological; any number of geometric forms may be derived from one by projecting it onto p space in different ways.

Moving down the levels of abstraction, we have seen that a precursor may be manifest[ed?] as an abstrract topology, or the abstraction generated. The abstraction is then realized, or the geometric form [concretised or projected?]. Moving up the other way, we can say that a geometric polytope may be abstracted or the abstraction drawn, then the abstraction sublimated or the precursor distilled.

[There are two levels of geometry - morphic and concrete. The concrete is formed by mapping or injecting the morphic p-tope into some p-space. Is this morphic space the one from whence the vertex figures are mapped?]

So we have the following representations:

Symmetrical precursors

Any geometric polygon may be represented by an anticlockwise circuit around its boundary, noting the sequence of corners and sides traversed. For exaple the polygon in Fig X is AeBfCgDh. When reciprocated with respect to the circle shown, another polygon a'E'b'F'c'G'd'H' is obtained, where a' is reciprocal to A, and so on. The prime marks ' denote the reciprocal relationship; omitting them we obtain aEbFcGdH. The representations for the two polygons are identical letter sequences, but with differing capitalisations. The capitalisation indicates which of the two reciprocal polygons we are dealing with; we may refer to the capitalisations as being reciprocal. The letter sequence, without any capitalisation, represents their schema: {{a b c d} {e f g h}}. Each polygon schema uniquely defines a dual pair; by adding capitalisation we denote the one polygon or the other.

Let us define some face of a polyhedron as AeBfCgDh and some vertex figure as JmKnLp. Some vertex figure of the dual polygon is then aBcDeFgH and some face is jMkNlP. Treating a polyhedron as two sets, of faces and vertex figures respectively, we find that the dual polyhedron is just the dual sets. Uncapitalised, these two sets form the schema of the polyhedron pair, with each member being a polygon schema.

I have adopted the convention that the circuit around a face starts at a vertex, whereas that around a vertex figure starts at a face - or, more strictly, the corner figure of a face. The two dual sequences can thus be distinguished in that the face starts with a capital letter, denoting a vertex, whereas the vertex figure starts with a lower-case letter, denoting a corner figure.

For regular or partially regular polyhedra, subsets of congruent faces or vertex figures may be identified using symmetry groups. For example A might denote all vertices within a given symmetry group. This allows considerable shortening of the notation.

For example the cube has vertices A and sides b. A face circuit is AbAbAbAb. Likewise, if it has faces C then the vertex figure circuit is cBcBcB. Knowing the relationship of these figures to the symmetry group, it is sufficient to identify them respectively as Ab and cB, denoting the cube as {{Ab} {cB}}. The regular octahedron is thus {{Cb} {aB}}.

This notation may be further condensed when studying the stellations of a particualr polyhedron and the facetings of its dual. A uniqe distinguishing feature of each stellation is its face diagram, and of each faceting is its vertex figure. These figures are reciprocal, so share the same schemas. We may ignore the other set of figures. For example when stellating the octahedron and faceting the cube, we would simply write Cb for the octahedron, cB for the cube, and cb for theor schema. For an example of the descriptive value of this notation, see the schemas of the stellated icosahedra and faceted dodecahedra [3 or 4?].

Where multiple separate non-congruent elements are present in the schema (for example, non-congruent faces lying in the same plane), they may be separated by a ~ (tilde) symbol. A set of multiple circuits connected by one or more ~ symbols may be called a compound schema and uniquely defines a dual pair of polygon sets. Examples of these are also to be found in [3 or 4?].

References

  1. B. Grünbaum, "Polyhedra with hollow faces", Proc of NATO-ASI Conference on Polytopes ... etc. ... (Toronto 1993), ed T. Bisztriczky et al, Kluwer Academic (1994) pp. 43-70.
  2. H.S.M. Coxeter, Regular Polytopes, Dover (1973), p.128.
  3. G. Inchbald, "Towards stellating the icosahedron and faceting the dodecahedron", Symmetry: Culture and Science Vol. 14, 1-4 (2000) pp. 269-291.
  4. G. Inchbald, "Icosahedral Precursors", a work in progress.
  5. S. De Haro and J. Butterfield; "On Symmetry and Duality", Synthese, Special Issue (Online), (2019). https://doi.org/10.1007/s11229-019-02258-x
  6. N.W. Johnson (ed. G. Inchbald); "Polytopes - Abstract and Real", steelpillow.com, (2008).
  7. P. McMullen & E. Schulte (2002). Abstract Regular Polytopes, CUP.