Fig 1 : A ditelon
by
Guy Inchbald
Updated 22 March 2005
This essay is unfinished. It needs reorganising before it can make much sense. In fact I've started another one which tells the story quite a different way. Who knows which will win.
A new definition of polytopes as set pairs is presented, based on the inclusion of vertex figures. Dual or reciprocal polytopes are understood as differing manifestations of the same precursor. A notation suited to the simpler and more regular cases is described.
22 March 2005. Precursors renamed from
'generators' (elsewhere I used yet another term, 'templates'). First half reworked.
The rest may never make it home.
Contents
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 yet 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 made up of sub-polytopes, or sub-topes; a line segment is bounded by points, a polygon by line segments, a polyhedron by polygons, a polychoron by polyhedra, and so on. But if we have no general definitions of polytope-ness or of how to fit polytopes together to define a higher-dimensional one, 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 semi-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 puts forward 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 elegant treatment of duality and reciprocation, 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.
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.
[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.
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.
I will try to provide both geometric and analytical definitions. On the analytical side, the language of set theory will prove convenient. This should not be taken to infer that a polytope is necessarily some mathematical set of entities, rather that such a set is just one way of describing it. In the following, Πp denotes some p-tope.
The null polytope is required by certain analytical formalisations, such as set theory. It has no geometric or topological significance. Every set has as a member the empty set { }. The empty set is for the present purpose identified with the null polytope, and has a dimensionality of -1 [ref ?].
A polytope in 0 dimensions is a point A. The only sub-tope (which might conceivably be available to correspond to any "facet" or "vertex figure") is the null polytope or empty set - that is to say, a 0-tope formally has no topological structure.
A 0-tope is defined as the set {{ } A} or, in abbreviated form, {A}:
Π0 = {A}.
A polytope in 1 dimension is a line segment, or ditelon. It is bounded by two 0-topes, {A} and {B}, which are its ends. The geometry at each end is defined by two associated end figures, {C} and {D}. These end figures may seem to be of trivial importance. However, they will become significant when building higher polytopes from such line segments.
Fig 1 : A ditelon
We can express this in set theory as:
Π1 = {{{A} {B}} {{C} {D}}} or, abbreviated, {{A B} {C D}}.
For now, the associations of C with A and of D with B may be made by the ordering of the set elements. Later on, we will see how this ordering may be discarded. [Answer - C=A' and D=B', but we don't find this out until the bit on duality.]
[There is no need for the ends to "connect up" in the same way as sides, so there is no need to make these sets closed under recursion. There is no link between A and B' or B and A'.]
[This is the first level at which the full definition of polytope applies. p=1 does not need recursion, see above.]
A polygon is made up of sides and end figures. The tetragon (or quadrilateral) in Fig 2 has four sides, {{A B} {E F}}, {{B C} {G H}}, {{C D} {J K}} and {{D A} {L M}}.
Fig 2 - A polygon (incomplete)
If the polygon had no corner figures, the corner angles would be undefined and the figure could be a rhomb of any angle. The corner figures are needed, to define the corner angles.
Note that as drawn, the corner figures are incomplete - they have ends but no end figures. For reasons which will become clear later on, their inverted end figures will be drawn. The lines of the corner figures may be extended to meet these inverted end figures, and, again for reasons which will become clear later on [I hope!], the end figures are found to coincide with the intersections of the extended lines. Thus two corner figures share each end section, as shown in Fig 3.
Fig 3 - The polygon completed
The corner figures may now be written as {{F G} {N P}},{{H J} {P Q}}, {{K L} {Q R}} and {{M E} {R N}}.
Each side has an element in common with each of two corner figures, and each corner figure has an element in common with each of two sides, in such a way as to link all members of both sets. Thus, every member of either set is linked to two members of the other set, wich in turn are linked to other members of the starting set. That is to say, the two sets are closed under this recursive linking. This recursion provides one-to-one connectivity of edges at their corners, resulting in a connected circuit around the figure.
Each element of the polygon is defined by the pair of adjacent elements of the other type: an edge is defined by the adjacent corner figures, a corner figure by the adjacent edges. Ordering information is contained within these recursive definitions of elements within the two sets. This recursion takes the place of the traditional abstract theoretical idea of semi-ordered sets.
[Use cube as the example.]
Fig
Recalling that a vertex figure refers to the surrounding faces and a face circuit refers to the surrounding vertices, we see that the two sets of polytopes provide recursive connectivity information: each element in one set contains references to elements in the other set.
A polyhedron may now be seen as an irreducible closed set of recursively connected polygons.
[Just make some general points.]
We can extend this indefinitely to provide a general definition: for p = 2 or more, a polytope in p dimensions is an irreducible set of recursively connected (p-1)-topes, with the set closed under recursion. Specifically, it comprises two subsets, with each member of each subset having (p-2)-topes in common with [members of the other subset... er, think about this.]
One of these sets is realized, the other remains [ideal/virtual?].
For p < 1 this definition must be modified, as noted below.
The manner of recursive connection is that each member of each subset shares each (p-2)-tope in common with just one member of the other subset [but check my paper scribblings]. This ensures continuity and closure of the surface.
The sets must be irreducible under recursion, otherwise the figure would contain smaller p topes and so be a collection of such. However, a collection of p-topes appropriately arranged and sharing a common centre is sometimes referred to as a compound polytope.
This new view of a polytope may be compared to the traditional view of it as a closed set of p-1 dimensional facets. A p-tope is still seen as a closed set of (p-1)-topes, but specifically of two different subsets whose elements are recursively interconnected. The recursive references must be irreducible and closed, or the polytope will not close in p space. It is evident that for every reference made in some precursor a to some other precursor b, b makes a reverse reference to a. This would appear to lay to rest the old problem of defining the interconnectivity of the (p-1)-topes.
[Use cube vs. octahedron as the example]
Geometrical reciprocation about some p-sphere twins every polytope with a reciprocal figure.
In 0 dimensions, the 0-sphere is just the same point A as the polytope, so the polytope is identical to the 0-sphere and is self-dual.
Grünbaum distinguishes abstract and geometrical polygons [1]. He defines an abstract polygon as a circuit in a simple Eulerian graph: the polygon comprises a set of elements (vertices) and a set of element pairs (edges). The ordering information for the circuit is contained in the ordering of the edges. This definition does not lend itself easily to duality. To obtain the dual polygon, the elements must be transformed into pairs and the pairs into elements. The ordering information cannot be contained within the new elements, and must be transferred across to the new pairs. This is all rather contrived and awkward, yet still fails to fully describe the polygon. For a general polygon, the ordering of sides and corners is insufficient to define it accurately. One also needs to know the angle at each corner, for example to determine whether the polygon is regular. Grünbaum's definition does not directly provide such metric information: the angles must be calculated from the vertex coordinates. However one manipulates one of Grünbaum's polygons, for full analysis one must independently determine the structure of the dual polygon before applying the dual manipulation. [cut this down].
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.
The insights which these ideas provides are so profound and illuminating that I believe they may be used to resolve the many issues over whether certain figures are or are not polytopes. Any acceptable definition of a polytope must lead to the following as axiomatic: if a certain figure is a p-tope, then its dual is also a p-tope. This axiom has many interesting consequences, only one of which will be followed up here.
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.
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?].