Polytopes: Degeneracy and Untidiness

Updated 10 Feb 2019.

Abstract

This essay is unfinished and still very much a work in progress.

Abstract polytopes are introduced as precursors to geometric polytopes. Geometrical untidiness is distinguished from from structural or combinatorial degeneracy. Some types of untidiness and degeneracy are discussed. Features located at infinity can have two opposing images.

Change history

10 Feb 2019. Abstract theory linked in. Much revised terminology. Other improvements.
9 April 2005. Precursors renamed from generators. Odd bits improved and a note on a forthcoming essay added, as much to show that I am still making progress on the current topic as for any other reason.
21 March 2004. Split off from "Polytopes, duality and generators" (now renamed "Polytopes, duality and precursors", with some new material.

 
Contents

Introduction

The ideas presented in this essay are generally applicabe to higher polytopes, although the focus here is on three-dimensional polyhedra. For the reader concerned only with the familiar three dimensions, where the word "polytope" is encountered, you may substitute "polyhedron" with no loss of rigour. Conversely, the student of higher polytopes will readily be able to extend the principles described.

Despite over 2,000 years of academic study, mathematicians have yet to come up with a satisfactory understanding of certain properties of the more complicated polyhedra. Some broad problem areas include:

Any coherent understanding of polyhedra and polytopes must take on board a number of theoretical advances which have been made over the centuries. It is worth summarising some of the more important ideas here.

We understand a polytope to be made up of sub-polytopes or subtopes, also known as its elements. The polytope is its own maximal element. The simplest polytope, of 0 dimension, is a point. Next up, in 1 dimension, is the line segment or ditelon. It is bounded by two vertex points, its sub-dimensional elements. A polygon is bounbded by a chain of ditela, a polyhedron by a surface constructed of polygons, and so on.

Every polyhedron has an abstract combinatorial structure which describes the connectivity or incidences of its elements. Such an abstract polytope may be distinguished from its geometrical form: any given abstract polyhedron may be realized as a wide variety of geometric polyhedra, all having the same underlying form.

Geometrical reciprocation about some sphere twins every ordinary polyhedron with a reciprocal figure. Reciprocating the second figure results in the original. The combinatorial structures of the two figures are dual to each other. Reciprocating about a different sphere yields a different reciprocal figure, however all reciprocals of a given polyhedron share the same combinatorial structure. A vertex of one polyhedron is dual to some face of the reciprocal. If a vertex figure of one twin is projected onto the plane of the dual face, the coplanar figures are found to be planar reciprocals. More about vertex figures of this and other types may be found here.

The theories of abstract duality and geometric reciprocation are quite general and may be applied to polytopes in any number of dimensions. The insights which they provides are so profound and illuminating that they may be used to resolve many of the issues over whether certain figures are or are not polytopes. Abstractly, it is a theorem that if a certain figure is a realization of a polytope, then its reciprocals are also realizations of polytopes.

Abstract precursors

Formally, an abstract polytope is a set of members or elements which are partially ordered by ranking, together with a certain incidence relation. Elements of a given dimension (point, lines, regions, etc) are given the same rank and any given element is incident with one or more elements in adjacent ranks. This partial ordering defines the combinatorial connectivity of the elements, and hence the particular abstract polytope.

As with all sets, the abstract polytope is completed by the inclusion of the empty set, which we may think of as the null polytope or nullitope. It has dimension −1. Thus, the set is "topped and tailed" by itself, the maximal element, and by the nullitope, the null element.

A remarkable property of such partially-ordered sets is that the dual polytope is obtained simply by reversing the ranking by dimension. For example the maximal element and null element swap dimesionality. In a polyhedron the vertices and faces swap their dimension, the vertices gaining the dimension of faces and the faces being reduced to points. But in the pure abstract expression, the direction of dimensionality need not be defined. Whether the set represents a given polyhedron or its dual is not determined until the figure is realized in some geometrical space.

In this way the abstract polytope can be thought of as a single precursor, which may manifest as either of two dual geometric polyhedra.

An elaboration of abstract theory is the selection of all the other elements associated with some given element to define a sub-polytope, which is called a section of the abstract figure. For example the particular section associated with a vertex is its vertex figure. We get into niceties over whether say a "face" is just the plane region or also includes its bounding sides and corners, but I will pass over such complexities here; the theory is rigorous, though the following discussion is not.

The abstract precursor thus gives equal importance to the faces and vertex figures in describing a polyhedron. It is immediately obvious that if a face is a polytope then so is the dual vertex figure, and vice versa.

The two subsets of these sections, faces and vertex figures, contain every element of the polyhedron and are sufficint to define its overall connectivity. To obtain the dual polyhedron, one simply exchanges the dimensionalities of the two subsets. The precursor makes these subsets explicit. For more about this, see Polytopes, Duality and Precursors.

Untidiness and degeneracy

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 on the one hand, and the corners and sides of a polygon on the other.

The term "polyhedron" has been applied to a wide variety of three-dimensional forms. A polyhedron-like structure which does not conform fully to the more standard definitions is sometimes said to be degenerate. However, different authors' ideas of what is standard and what is degenerate tend to differ. This difference is partly due to their differing areas of interest, and partly due to a failure to separate geometric and topological properties. That is to say, traditionally a degenerate structure may have a degenerate topology, or a well-formed topology but degenerate geometry. Notice that if the topology is degenerate, the geometry will necessariliy be so too.

Johnson noted this difficulty and proposed that "degenerate" should be reserved for abstract forms which are no longer polytopes, for whatever reason. For a malformed geometric realization of a true abstract polytope, he suggested the term "reductive" (see here). But some malformations, such as curved elements, are not really reductive, and so I use the term "untidy" to mean the same thing.

Both Johnson and I have independently made the point that ideas of what is or is not reductive or untidy vary widely depending on the particular geometer's area of interest, and that it is therefore a key aspect of the term that its exact meaning is subjectively defined or, at best, defined locally within the specific context under discussion.

Some common examples in Euclidean space of polyhedra which we might think of as untidy include:

Curiously, while many authors have described polyhedra having one or more of these untidy properties, a planar figure exhibiting any equivalent properties is seldom accepted as a polygon. For example a quadilateral with curved elements may be accepted as a polygon when drawn on a sphere but not when drawn in the plane. There is a lack of consistency in this tradition, which requires cleaning up.

Coincident features

If a polyhedron has two colinear but separate edges, its geometric reciprocal may have these edges superimposed such that four faces meet at the combined edge. For a geometric polyhedron this is traditionally not allowed, or at best regarded as degenerate (though there is of course some debate). Note however that combinatorially it will retain the two distinct edges of the original, and remains well-formed. Consequently we may now understand such forms as Skilling's Great disnub dirhombidodecahedron to be genuine polyhedra, albeit having an untidy appearance.

Features at infinity

In Euclidean space, if a face plane passes through the centre of the reciprocating sphere then the face reciprocates to a vertex located at infinity, i.e. one which has no place in Euclidean space. The incident vertices reciprocate to infinite prisms. Combinatorial duals of such "hemi" forms, and even geometric reciprocals in elliptic or projective space, cause no such problems.

The hemi faceted cube has six "butterfly" cross-quadrilateral faces and three rectangular hemi faces. The crossing points in the middle of each cross-quadriateral are "false vertces," i.e. there is only the illusion of a vertex. This cube is unusual in that its vertices are chiral, with dextro and laevo forms alternating.

 
Fig : Hemi faceted cube - 3D view and vertex figure

The reciprocal of the hemi faceted cube is an infinite stellation of the octahedron. I will call it an infinite octahedron. It has the appearance of six rhombic prisms, extending to infinity. Each face is bounded by three pairs of parallel edges, each from a different prism.

The traditional view is that the prisms extend to infinity in only one direction, which may be chosen arbitrarily. However, it is not possible to make a choice which both closes (i.e. fully bounds) the polyhedron and preserves both congruency of the infinite faces and symmetry of the polyhedron - conditions which the reciprocal must surely meet - so any such choice seems unacceptable.


Fig - Traditional mess

[3] presented an argument that a vertex at infinity has images in both directions. This provides the required closure, congruence and symmetry. It is unreasonable to treat different polyhedra differently on a whim, so all such prisms, reciprocal to any hemi faces, must have both ends.

In the centre of each infinite face is a triangular region, internal to all three pairs of prisms and having three vertices. At each of these vertices, three edges intersect - one from each pair. Each edge appears to be geometrically trisynaptic, having two finite vertices and a third at infinity along its length. This is untidy. The reciprocal vertex figure is both tidy and chiral, yet the geometric face appears to be neither. However, combinatorially each edge is connected to only one of the finite vertices. That is, a true vertex and a false vertex coincide. By disconnecting the correct edges, the trimethoric character is removed, and chirality restored. The edges and vertices are combinatorially tidy but appear geometrically untidy.


Fig - Face of the infinite octahedron

Chirality is generally treated as a geometric property, and not a combinatorial one. Yet we have seen that it does not appear to be preserved under reciprocation. How can this be? Either the form of chirality seen here is combinatorial in nature, or the meeting or missing of an edge passing through a vertex is a geometric property and not merely combinatorial. I have not yet figured the answer to this one. Possibly, this is because I do not understand the difference between projective and other kinds of geometry.

The reciprocal feature to the apparently trisynaptic edges is that along any edge of the original cube, three face planes (but not three faces) share a common line of intersection. We see the hemi solids as geometrically tidy, while their infinite reciprocals are untidy.

Holes

Elsewhere [3] I have mentioned the difficulty of allowing for certain kinds of hole in a polygon or polyhedron. A polygon essentially traces a circuit, which is not particularly useful here when things get complicated: holes can punch through overlapping circuits or loops and only really become significant when viewed from a higher dimension.

Holes are really just the flip side of filling the interior of a polygon or polyhedron. More about the importance of fillings can be found here.

Curved faces

Yet another class of geometrically untidy polyhedra are those with non-planar faces. Here, the edges of a face typically form a skew polygon, and may or may not be straight. Any combinatorial polyhedron may be represented by a simply-connected graph [2]. A circuit enclosing any region of the graph represents a face - a circuit around an irreducible region represents a face of the outer hull, and a wider circuit around several such regions represents an edge facet of that hull. Combinatorially this is all nice and tidy, but geometrically the general circuit is skew, leading to skew facets; it is only in certain special cases that the circuit and facet are planar.

The skew facets may be thought of in different ways, as empty in the manner of Grünbaum, as filled with a curved minimal surface such as found in certain theoretical bubbles or saddle polyhedra [4], or as filled with some other specialised surface such as the walls between real bubbles of different sizes, in a foam. Bubbles tend also to have curved edges. One may also observe that there are a large number of facetings of a typical polyhedron, of which only a relatively small subset have planar faces. The question arises as to whether the non-planar cases reciprocate to any meaningful polyhedra. If a general curve is treated as a string of arbitrarily short line segments joined by vertices, then its reciprocal is seen to be a string of vertices joined by arbitrarily short line segments, which is a similar construction, viz. another curve. [No doubt the theory of reciprocal curved lines and planes goes a lot further. If reciprocation does break down, how do we account for the dual structure? If not, what is the reciprocal of a bubble?]

Some degenerate structures

Candidates for degenerate polyhedra in Euclidean space include:

Tilings are sometimes seen as infinite polytopes, or apeirotopes, of higher dimension, having an infinite number of faces. For example plane tilings are apeirohedra. Certain non-planar arrangements also extend infinitely, dividing space into two parts. They are also apeirohedra, and have been variously called infinite polyhedra [Coxeter], honeycombs [], sponges [] and pseudopolyhedra [Gott]. How can they be polyhedra if they have no interior? Their structures conform to the definition of an abstract polytope, which allows infinite sets, but their maximal element is not realizable as any kind of interior. Because of this Johnson called them "improper" realizations, within the context of his own theory of "real" polytopes.

Any attempt to construct the regular star polyhedron {5/2 5/2} in Euclidean space goes on endlessly, with unpaired edges of polygons always present. It is in effect an infinite polyhedron much like a tiling, although it is tiling the sphere an infinite number of times rather than a flat plane only once.

But not all infinite polyhedra are degenerate. The Spidron polyhedra of Daniel Erdely have faces spiralling sequentially towards a certain vertex in ever-decreasing size, requiring an infinite number of ever-smaller faces to reach the vertex. Nevertheless the polyhedon conforms to abstract definitions and bounds a finite interior. These polyhedra are examples of where the abstract precursor is infinite but the realization is not. Hyperbolic tilings realized as a Poincaré disc provide further examples.

In many complicated self-intersecting figures, such as may be creted through stellation, we can readily find examples where faces or vertices are visited by multiple circuits, or edges are visited twice by the same circuit or by different circuits. In certain situations, in a given surface n-space, we may find a compound or a composite where the figure divides up into structurally discrete but geometrically overlapping polyhedra. Such a compound is structurally degenerate, as it comprises multiple realizations of one or more abstract polytopes.

Some geometric figures may best be seen as n-methoric or n-synaptic forms, having n faces or vertices along a given edge. The structure of such figures breaks the dyadic rule, that only two (n-1)-topes meet along any (n-2) tope, so these figures are degenerate. The principles of duality still apply, so precursors remain a useful way of describing them. Indeed, since simple circuits around faces or vertex figures are no longer possible, precursors are one of the few approaches which do remain of value.

In all these cases, the dual form is also degenerate. We can say that in general, the dual of a degenerate polyhedron will also be degenerate.

Conclusion

I have previously suggested [3] that the dual or reciprocal of a tidy polyhedron should also be tidy. However, now that we have made the distinction between degeneracy and untidiness, we can say that while the dual of a degenerate structure will also be degenerate, the reciprocal of a tidy polyhedron may be tidy or untidy depending on the chosen projection (i.e. the chosen realisation and/or the chosen sphere of reciprocation).

Whe have seen how different geometrical figures can be classified according to their degree of degeneracy. For example the infinite "hemi duals" are combinatorially well-formed, and result from an accident of sphere location which has nothing to do with their underlying form. It is to be hoped that this classification will help bring some kind of order when asking of various figures, "is this a polyhedron or not, and if so then what kind is it?"

References

  1. N. Johnson; "Polytopes - abstract and real.
  2. Skilling,
  3. G. Inchbald, Towards stellating the icosahedron and faceting the dodecahedron , Symmetry: Culture and Science Vol. 14, 1-4 (2000) pp. 269-291.
  4. P. Pearce, Structure in nature is a strategy for design, MIT (1978).
  5. Gott.