Polytopes, duality and precursors

Updated 15 Apr 2019

The idea of precursors had its roots many years ago in the duality of stellation and facetting. It has evolved substantially over time and this study has been almost wholly rewritten and renamed from its original form. 'Precursors' were once called 'generators' or 'templates' but for technical reasosn neither term proved satisfactory. The mathematical foundation turned out to be intimately linked to modern abstract theory and, in the recasting of it into abstract concepts, I discovered significant flaws. Despite a couple of major revisits to date, precursors remain a work unfinished and parts of this essay have yet to be updated.

Abstract

The duality of polyhedra is inherent in both the geometric process of reciprocation and the set-theoretic formulation of abstract polytopes. In order to construct a given geometric figure from its individual elements, the geometries of both faces and vertex figures are required. A geometric vertex figure is a realization of an abstract vertex star. The twin sets of the individual faces and vertex stars comprise a common generator for both a geometric polyhedron and its dual.

Contents

Introduction

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, in one dimension a line segment, in two a polygon, three a polyhedron, four a polychoron, 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.

Despite over two millennia of academic study, mathematicians have struggled to come up with a satisfactory formal definition of polytopes. Grünbaum referred to this failure as "The original sin in the theory of polyhedra" [1]. Nowadays we generally understand a polytope to be a geometrical "realization", an injection into some parent space, of a particular kind of set-theoretic structure known as an abstract polytope. The polytope comprises "elements" which are partially-ordered in ranks, with a certain pattern of incidence relations defined between them. Elements of rank 1 are vertices, of rank 2 sides, and so on. But if we have no rigorous definition of the realization process, then we have little idea what geometrical properties the resulting polytope might have, and this is not much of an advance.

See for example Coxeter [2] who built his analysis on that of Schläfli. Grünbaum tried to formalise this analytically by organising the subtopes into partially-ordered sets [1]. Since then he and others have further refined and generalised this model of "abstract polytopes", to the point that they need no longer bear any resemblance to the real thing. It seems to me that this is missing the point of what a theoretical model is for: we want it to inform us about something that is real, and help us to distinguish objects that can exist from those that cannot. OK, only three dimensions of space exist, but the idea of higher-dimensional polytopes would, as it were, be valid if suitable spaces did exist, and we want to know about the beasts that could inhabit such spaces rather than ones that couldn't. Whatever those elegantly generalised abstract structures are, they no longer fit the label "polytopes". We still lack a model which accurately matches the properties of the real thing.

This essay considers such a new model, based on the need to define the structure and geometry of vertices as well as faces. As presented here, development of this idea leads first to an aproach to duality and reciprocation based on the duality of polytopes in abstract theory, and then on to a level of order underlying even the topology of the forms.

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

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.

A polytope having one dimension less than the encompassing space is a (p-1)-tope. A (p-1)-tope which bounds a p-tope is generally a facet. A facet in one dimension is also an end, in two a side, in three a face, in four a cell, generally in j dimensios it is a j-facet. The idea of bounding is deliberately left vague here, as it opens some difficult and unresolved issues over holes. I hope to address these at some other time. The reader's intuitive understanding should be sufficient for the present purpose.

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 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.

One way to provide this extra 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. 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 readily 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, 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.

Coxeter [2], presumably still following Schläfli [check this], defines a regular polytope as one having regular facets [did I read "faces"? - check] and regular vertex figures. It is a short step to say that such a polytope is defined by both these sets of figures together, and more generally that any given polytope is defined by both sets.

We now modify our hierarchical definitons of polytopes accordingly: a 3D polyhedron is now made up of 2D faces and vertex figures. These in turn comprise 1D sides and corner figures. This hierarchy may be extended generally to p dimensions, for example a polychoron (4-tope) comprises sets of polyhedral cells and polyhedral vertex figures.

More about vertex figures

[Depending on the exact definition of the vertex figure, we may distinguish intrinsic and polar types. The geometry of an intrinsic type depends solely on that of the vertex. A polar vertex figure occurs as the reciprocal figure to the face of the reciprocal polyhedron; its geometry depends also on that of the associated reciprocating sphere. [see xref].]

[Notice that in order to define the geometry at a vertex, we do not have to use the interior angle of the polytope. Any angle included between the intersecting sub-spaces will do, provided we remember which one. [comment on the problem of similar vertex figures for different "interiors" of the same vertex, e.g. overhanging vs. congruent vs. saddle.]

This essay will often use the inverted vertex figure, which occurs in the angle opposing the interior angle. For example in one dimension the end figure lies a short distance from the end, whereas the inverted end figure lies a short distance beyond the end, on the extended line of the segment.

Abstract polytopes

We will recognise the need to include information about vertices by defining a polytope as a collection not only of its facets but also of its vertex figures.

My theory of morphic polytopes provides a useful approach to understanding. On the analytical side, the language of set theory will prove convenient. The various elements of a polytope - its faces, edges, vertices and so on - may be collected together to form a set. Elements of a given type, such as all faces, are ranked together, producin a partial ordering of the set. The adjacencies or connectivities of elements of different rank, such as the sides of faces, form an incidence relation which further defines the partial ordering. For completness in set theory, an element of maximal rank is added and the empty set is added as the element of minimal rank. This partially-ordered set is an abstract polytope.

Realization

The above should not be taken to imply that a polytope is necessarily some algebraic set of entities, rather that such a set is just one way of describing its structure. A geometric figure in say Euclidean space must be derived from the abstract formulation through some process of realization.

Morphic theory recognizes three steps or stages necessary to realize a geometric figure, of which two are significant here:

Firstly we define 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 empty set is left as a kind of null figure. The incidence relation is defined as a structural adjacency or connectivity.

The assemblage of all these elements now comprises a topological manifold, a "rubber-sheet" figure. Morphic theory treats this figure as a bounded manifold, the boundary being the polyhedral surface and the rest of the manifold its interior. This contrasts with the more traditional twentieth-century focus on the bounding surface.

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

Duality

Abstract duality

A polytope of n dimensions has two more ranks than its number of dimensions. In the first realization step the maximal element is given rank n, the next down n−1 and so on down to vertex points with rank 0. The empty set, ranked below 0, therefore has rank −1 and the total number of ranks is n+2.

It is a remarkable property of such realizations that the dual polytope is obtained simply by reversing the order of ranking. The element that had been maximal now becomes the null entity and vice versa, and so on all the way through. For example the vertices and faces of a polyhedron become respectively the faces and vertices of the dual.

This, the same abstract polytope underlies both figures, a polytope and its dual. It is only in the first step of realization that the choice is made as to which of the dual geometric figures is to be realized.

Duality and set pairs

[Use cube vs. octahedron as the example]

Geometrical reciprocation about some (n−1)-sphere (for n≥1) twins every n-polytope with a reciprocal figure. (for n=0 the (−1)-sphere is undefined).

In extending the idea of set pairs to polyhedra, we make use of the well-known property that the dual of a face is a vertex figure of the dual polyhedron, and the dual of a vertex figure is a face of the dual polyhedron. A corner figure is a side in the vertex figure, and an edge section is a corner in the figure.


Fig 2 - A polyhedron and its reciprocal

Any cell of a polytope is dual to a particular polar vertex figure of the reciprocal polytope. For example, any face of a polyhedron is dual to a particular polar vertex figure of the reciprocal polyhedron.

[Connectivity of VF's - a crucial bit of magic, if it works out right!]

We can now see that any defining sub-polytope, or sub-tope, is dual to some defining sub-tope of the dual polytope.

In this picture not only do cells and vertex figures have equal status, but they have an intimate relationship which will be formalised in my definition of a polytope. Note the equivalence of set pairs here.

For p=1 a line segment AB may be reciprocated about a 1 sphere (another line segment) to yield the line segment A'B'. [Revisit 1-topes to show that C=A' and D=B'.]

One may distinguish a polytope's reciprocal, which is the geometric polytope obtained when it is reciprocated with respect to a given sphere, from its dual, which is its topological opposite number.

Precursors

Realizing vertex figures

A key distinction is made between a single element, such as a face, and the subset including elements of lower dimension which form its boundary. These bounding elements, together with the bounded element and the empty set, form a polytope in its own right, although of lower dimesionality. For example a four-sided polygonal face has four bounding sides and four bounding vertices, with the complete subset forming an abstract polygon. In abstract theory this bounded polygon is often called a 2-facet while the original element, its body, is a 2-face. This terminology is unhelpful if one is concerned with "faceting" as reciprocal to stellation, so here I adopt Johnson's use of "facial" to denote the sub-polytope. For a face of rank n−1, the elements of the associated facial have ranks −1 to n−1.

A vertex star is the set of elements which are incident on a given vertex. Its ranks within the parent have dimensions 0 to n. It has the same number of ranks as an n−1 facial, but shifted up one rank. When the polytope is dualised, the elements of a vertex star become the elements of the dual facial, and vice versa.

A vertex figure of the same vertex has exactly the same incidence structure as the vertex star. Geometrically it may be understood as a slice or section through the star, so consequently its dimensionality is lower. For example a face in the vertex star becomes an edge of the vertex figure, while an edge in the vertex star becomes a vertex of the vertex figure. A vertex figure can thus be understood as a sub-dimensional realization of the vertex star, shifted down one rank. This gives it the same absolute ranking or dimensionality, of −1 to n−1, as an n−1 facet.

Precursor sets

The set of n−1 dimensional sections of an abstract polytope comprise two subsets, its facets and its vertex stars. Which subset is which depends only on the direction or orientation in which its ranks are ordered. I shall call the as-yet unoriented set of sections a precursor.

When the polytope is fully realized as a gemetrical figure, the geometry of each face or facet 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, simply given the intrinsic geometries of each vertex figure is insufficient. For example they are identical for every cuboid, so these figures 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 precursor. Only in this way can the coordinates of each part of the figure be uniquely determined and the injection of the morphic manifold into metric space be fully defined.

It is this definition of the particular injection which has proved so elusive a corollary to abstract theory. Although I have presented precursors in abstract terms there is no strict need to do so and it would have been possible to give a purely geometrical account. However the importance of abstract theory makes a discussion in the context of that theory more useful.

Definition of a geometric polytope

It then becomes natural to actually define the geometric figure in terms of its precursors.

Polytope precursors

Earlier we defined a polygon in terms of its sides and corner figures. The length of the corner figure determined the angle at the corner (vertex). We then found that the face and vertex figure subsets have the same structure, and are joined in a symmetric arrangement. If presented with a given subset out of context, we would be unable to say whether it comprised the faces or vertex figures of a polyhedron. Indeed, if presented with both subsets out of context, we would be unable to say which was which. Before we can define a polyhedron, we must decide which subset to realize as the faces, and which as the vertex figures.

But wait, what is this set of sub-topes, if it has not yet been realized as a polytope? I will call it their precursor. The precursor represents some abstract mathematical entity, and the two polygons (topological or geometric) are dual manifestations of it.

To obtain the dual polygon, one simply exchanges the topological realizations (sides or corner figures) of the two sets. A precursor is thus a higher form of abstraction, from which either of two dual abstract polygons may be derived.

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

p=0 is a degenerate case which is a single point. 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 precursors {{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 - ideal and concrete. The concrete is formed by projecting the ideal p-tope into some p-space. This cannot be a graph in (p-1)-space, because the graph is not metric and is not a polyhedron. Is this ideal space the one from whence the vertex figures are projected?]

So we have the following representations:

I have not proved that any given precursor necessarily results in a unique pair of topologies: it may be that the notation or the underlying ideas need modifying to ensure uniqueness, especially when applied to more complicated polytopes. It should perhaps also be noted that the precursor as expounded here may not be relevant to all situations. For example it may not be applicable to certain kinds of hole.

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 precursor: {{a b c d} {e f g h}}. Each polygon precursor 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 precursor of the polyhedron pair, with each member being a polygon precursor.

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 precursors. 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 precursor. For an example of the descriptive value of this notation, see the precursors of the stellated icosahedra and faceted dodecahedra [3 or 4?].

Where multiple separate non-congruent elements are present in the precursor (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 precursor 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)
  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.