Updated 16 Feb 2008
In polytope theory, filling is shown to be of fundamental importance. Traditional theory ignores filling, and so is incomplete. This essay is incomplete too.
16 Feb 2008. New Figs cuploid.gif and keratinoid.gif, Grünbaum's proof, Minor additions and tidyings.
The basic idea of polygons and polyhedra may be generalised in other sets of dimensions as polytopes. A polygon is a polytope in two dimensions, or 2-tope, and a polyhedron is a 3-tope. The reader who is only familiar with the usual three dimensions should be able to understand this essay, while students of higher dimensionalities should also find it relevant. In higher dimensions, a 4-tope is a polycell or polychoron, and so on. In lower dimensions, a 1-tope is a closed line segment or ditelon, and a 0-tope is a point or monon.
In general, we say that a polytope in a given dimension is built up from polytopes of the next lower dimension. For example we say that a polyhedron is built up from polygons, which form its faces.
Traditionally, the filling of polytopes has not been thought to have much significance. Different researchers made different assumptions about whether their polytopes were solid or empty, and it seldom seemed to matter much at the time. But when one looks carefully, one finds yet another fundamental muddle to be sorted out - so fundamental in fact, that I now believe that the proper description of any polytope must include its filling.
So why does it make a difference whether a polygon is filled or hollow? A polygon unambiguously divides the plane into inside (smaller) and outside (larger) regions, which is enough for most people. Let us use some polygons to construct a polyhedron. Six filled squares make a closed cube, or six hollow squares make a skeletal cube. Both of these cubes have very respectable credentials. Now, can we mix open and closed squares to make an open box? Five filled squares plus one hollow square make our open box - this is a little messy, but all six sides are perfectly respectable polygons, so why not? Fans of filled polygons would say that the open end is not a side so the open box is not a polyhedron, but are they right? Fans of skeletal polygons would say that the fillings are an irrelevant afterthought so the open box is (or at least incorporates) a cube and so is just as much a polyhedron as its skeletal sibling, but are they right? To get messier, are these six sides all the same kind of polygon or not? They are all squares having the same structural arrangement of edges and vertices, and so traditionally all have the same mathematical definition. But one square is obviously different in some way for the box to be open on that side. We are forced to conclude that traditional definitions are insufficient; in describing these squares we must also consider whether they are filled or not.
How has this shortcoming in polytope theory been missed for so long? I believe that it comes down to the way we visualise polytopes. Figure 1(a) shows a star pentagon, and 1(b) a decagon having the same external form. A two-dimensional flatlander standing next to them would not be able to tell them apart and might well consider the distinction trivial, yet to us three-dimensional mathematicians the difference is obvious. 1(c) shows a star polyhedron. Is it constructed from 12 star pentagons like 1(a), 12 nonconvex decagons like 1(b) or even 60 isosceles triangles? Now it is us who cannot tell the difference, yet to a four-dimensional demon it would be obvious.
Figure 1 : Star pentagon, nonconvex decagon and small stellated dodecahedron
Figure 2 shows three ways to fill a cube. In 2(a), just the 0-topes (vertices), have been filled in. The diagram is not very instructive, in fact it might even be showing a stella octangula (a regular compound of two tetrahedra) or any other figure having the same vertices. 2(b) and 2(c) are much more understandable, and both are commonly used.
Figure 2 : Three different fillings of the cube
Grünbaum in  was only concerned with 2(b), defining a face as the set of surrounding edges. He ignored fillings such as 2(c). One could equally well also ignore fillings such as 2(b), define an edge as the set of bounding vertices as in 2(a), and develop a comprehensive theory of such polyhedra. Nobody has bothered, partly because the limitations of such a theory are rather more clear, and partly because the diagrams would be both dull and uninformative. Our four-dimensional demon would most likely level the same criticisms at Grünbaum's hollow figures.
On the other hand, when Coxeter et. al. analysed the stellations of the icosahedron , they treated the resulting forms as solid lumps. As a result they recognised neither internal structure nor the falseness of certain vertices and edges where cells touched, and their selection of 57 such icosahedra is well-nigh meaningless .
There is already a problem with 2(c) in that we cannot tell whether it is an empty box or a solid. Traditionally we have not worried about this. But now look at Figure 3, which shows three fillings of a tesseract or hypercube, a 4-tope constructed from eight cubes. Figure 3(a) shows only the vertices and is almost useless. Figure 3(b) shows the edges and is much better; our eyes certainly need those edges filling in, don't they! Here we can immediately see the general structure, and with a little study we can discern the eight individual cubes, distorted to varying degrees by the projection from 4-space to-2 space. Figure 3(c) is now less helpful than before; not only can we not tell whether the cubes are filled or if the 4-tope itself is solid, but the near faces have obscured much of the polytope's structure from view. So students of the higher dimensional polytopes often stick to edge-fillings only.
Figure 3 : Three different fillings of the tesseract
A feature of computer simulations reinforces this habit. Computers are very useful for manipulating higher-dimensional polytopes, because they can be easily programmed to cope with the extra dimensions. When it comes to displaying the polytope, it is relatively easy to program the drawing of lines between the vertices, but much harder to program where solid faces are visible and where they obscure each other, never mind finding false edges where faces of a nonconvex polytope intersect. So we have tended to stick with what is both easiest to program and clearest to look at, and just draw the edges.
So you see, which n-topes we traditionally fill in depends very much on how we like to visualise them. Our eyes need filling of ditela most of all, fillings of polygons and polyhedra come and go according to our line of study, but above that we very rarely fill things in. This filling of polytopes according to the way our senses work is not mathematically rigorous. In fact, it's amazing that theoreticians have gone along with it for so long. We need something better.
Here's a more subtle problem. Traditional polygon theory does not concern itself with certain kinds of hole. Regions surrounded by the bounding edges are defined as being in the interior, there is no distinction as to whether a region is filled or not. As a result the theory is not able to describe certain kinds of polyhedral structure, such as the hole in Figure 4(b).
Figure 4 : Fillings of a star pentagon
Consider the two star pentagons in Figure 4. Figure 4(a) shows the filling usually given by mathematicians, which distinguishes interior and exterior regions. The central region is said to have a density of 2 because the boundary winds round it twice. Seems reasonable? Well, many general-purpose graphical drawing programs for computers will give you Figure 4(b). Programmers seem drawn to the logic that a boundary is a division between inside and outside, no matter where it occurs. This time, mathematicians and programmers take opposing views.
It is also possible to find polygons naturally having the form of Figure 4(b). Figure 5 shows two views of a polyhedron with pentagonal pyramid symmetry, having five kite faces and five cross-quadrilaterals. Although this example is rather flat, the general shape resembles a hollow horn, so we may call it a keratinoid. The vertex figure of the central apex is a pentagram whose centre lies outside the polyhedron, and so is hollow.
Figure 5 : Pentagrammatic keratinoid
Figure 6 shows two equivalent views of the dual polyhedron, the star pentagonal cuploid. This polyhedron is usually depicted with a filled-in pentagram, as here. The central region is in fact a membrane separating the inside and outside of the "cup". This is usually justified on the grounds that the region has denisity 2 and so acts as a kind of double-skin. This led Cayley to realise that the core of the small stellated dodecahedron has density 3: once for the surface of the outer spike, and twice for the centre of the pentagram surrounding the core. The densities of surfaces and of the regions of space beneath them cannot be treated separately. Returning to the cuploid, notice that the regions of space on either side of the membrane both have density 0. This must mean that the membrane also has density zero, otherwise the densities either side would be different, after the manner of Cayley.
Figure 6 : Pentagrammatic cuploid (incorrectly drawn)
In order to understand how this can happen, consider the fact that the polyhedral surface is non-orientable: that is to say, we cannot separate out an "inner" side from an "outer" one. Try it - start on one side, say the outside, and by tracing up over the rim and down again it is possible to end up on the inside. In these circumstances, it is not possible to assign densities to deep regions of the interior in the way that one can for, say, the regular star polyhedra - the best we can do is to assign a parity (0 or 1 modulo 2) to each region. Branko Grünbaum sent me the following explanation:
Let P be a polygon in the plane. At this point let's think of it as a bunch of segments. If you insist on "sides" being just pairs of points, one needs to adjust the language, but not the idea. Let L be the union of the lines determined by the sides of P (disregard the ones of zero length). The complement of L in the plane is a finite family of open regions R1, R2, ... , Rk. Take a point V in one of these, and a ray H that issues from V and does not go through any vertex of P (thus only finitely many rays are disallowed), and count how many sides (edges) of P are crossed by H. The parity of that number determines whether d(V;H) , the density at V with respect to H, is 0 or 1. Next, observe that d(V,H) does not depend on H, by considering what happens when you rotate H about V: Most of the time the same sided are crossed, but when H passes a vertex some or all sides can be changed and their number as well -- but all leaves the parity (that is, d(V,H) ) unchanged. Call that value d(V). Next, observe that d(V) does not depend on V as long as we stay in the same region Rj, since each of these region is convex and we can take the ray that starts at one point and goes through the other to see that d(V) is the same for them. Finally, let Ri and Rj be regions that have a common segment in their boundaries. Consider two points, one in each region, such that the segment connecting them intersects the segment separating them. When proceeding along a ray from one of the points to the other, the parity of d(V) will stay the same or change according to the parity of the number of edges that have to be crossed (the segment may belong to several edges of the polygon).
Now that we know what happens in the plane, we can go to polyhedra. Consider the complement of the union of planes that contain the faces, and for any point V in the complement determine d(V) as before, but for each point at which the ray crosses the plane of a face, observe what is the parity of that point w.r.t. the face. Since any two rays starting at the same point are in a plane, the intersection of that plane with the polyhedron reverts us to the earlier situation on noticing that d(V) is the same whether calculated as stated in the preceding sentence, or in the preceding paragraph. And that's all. Clearly, one can go up in dimensions. Also, clearly, in some cases it may be possible to give sharper counts than just the parity -- this certainly happens with orientable polyhedra.
In other words, densities higher than 1 are the exception, not the rule. Unlike orientable polyhedra, there is no sound mathematical reason to make the cuploid an exception.
Having seen that our partly-filled pentagram is theoretically watertight, how are we to describe it, so as to differentiate it from the fully-filled one? We cannot add more vertices around the pentagonal central hole, for this would drastically change the figure - making it a set of five vertex-connected triangles, which is not what we meant. If we say that we now have six polygons - five triangles and a pentagon - then we have the kind of mix 'n' match filling we met earlier with the open box. If we say that the hole has no vertices, then conventional theory has no way to describe it, though computer programmers evidently have. If you don't believe me, make a star pentagon from five crossed sticks joined only at their ends. Now glue on five paper triangles, leaving the central hole empty. Have you changed the sticks? No, they are the same star pentagon as before. Using any one conventional definition of a polygon, can you give a consistent description of the figure in your hand? It seems not - we will have to find out how the programmers do it.
Before we do so, let us look at one last example. Figure 7 shows stellation f1 of the icosahedron . Each face is bounded by two overlapping regular triangles. The equivalent polyhedron having fully filled faces is the well-known regular compound of ten tetrahedra, but here the central region is empty. By failing to fill this central region of each two-dimensional face, the topology of the three-dimensional figure is drastically changed and it becomes a toroid of genus 11.
|Figure 7 : Stellation f1 of the icosahedron and its face diagram|
A prison makes a nice analogy. Traditionally, a prisoner is held captive whether they are inside a building or in the open air such as in a walled compound; whether the area is roofed is not significant, they cannot climb over the walls. Consider a prison built to the plan of Figure 7(b), having six cell blocks around a central compound: no prisoner can escape from this compound. But now suppose that a friendly helicopter flies over and lowers a ladder - a prisoner in the compound can climb into the third dimension and escape, while a prisoner in their cell can not.
Returning to the stellation, the two regular triangles bounding each face have six vertices in all. The central hole has no vertices, and as such cannot be described by traditional theory. Adding vertices at each crossing point would destroy the elegant simplicity and symmetry of the faces, and would also result in a very different structure.
We can now understand that when a two-dimensional polygon (or set of overlapping polygons, as in this example) is used as the face of a three-dimensional polyhedron, its filling has a fundamental significance. Traditional theory cannot describe the difference between the faces, and so cannot account for the differing topologies: because it ignores fillings and holes, it is simply not up to the task of describing such polyhedra.
To find a more rigorous answer, it is best to start at the simple end. A monon or 0-tope is a point which has been identifed as a discrete geometric object. A point which is not recognised as discrete is not a proper polytope - it is merely a point in space. Likewise the idea of an empty monon has no significance - it is again just a point in space. We may equate the identification of a monon with the property that it is filled.
A ditelon or 1-tope with a filling is the familiar closed line segment, comprising two monons (ends) together with the filling (open line segment). A ditelon with no filling is just the two end points, which divide some line into inside and outside - or do they? In some geometries a line runs back on itself rather like a circle, as in Figure 8: a single point does not divide the line, and while two points do divide it into two segments there is no way to tell which segment might be "inside" and which "outside" the point pair. We need something extra to identify which segment is the interior of the ditelon.
Figure 8 : A line divided into two segments by two monons.
For the present purpose, the choice of segment is arbitrary. Having made our choice, we need to identify the chosen segment in some way. Otherwise, we will finish up with the ambiguous and unhelpful situation shown in Figure 9.
Figure 9 : Two monons bounding an empty ditelon.
(In some circumstances this problem can be so thorny that mathematicians despair and take the whole line to be the ditelon, giving up all thought of inside or outside. The end result is something called a configuration, and opinions differ as to whether such a thing can righly be called a polytope at all).
As with the monon, we can identify the interior geometrically by filling it, i.e. we equate the identification of a ditelon's interior with the property that it is filled. The ditelon AB in Figure 10 is quite clear.
Figure 10 : Two monons bounding a filled ditelon.
Recent years have seen the theory of abstract polytopes developed. An abstract polytope is a partially-ordered set (poset) of elements having certain properties. Any real geometric polytope may be described by mapping the various elements of the poset to the vertices, edges, etc. of the geometric figure. The figure is said to be a realization of the abstract polytope. If a one-to-one mapping can be made, the figure is said to be a faithful realization, and is a real polytope.
Such posets always have an element which corresponds to the body, or interior, of the figure. The question of filling such a figure is now that of whether we may successfully map the abstract body element to the geometric interior of the figure.
For a monon, the vertex element is also the body element. If we map the vertex element to some geometrical point, we find that we have also mapped the body: a monon is always filled.
For a ditelon, the edge element is the body element. If we map only the vertex elements to a point pair, leaving the body element unmapped, the figure is unfaithful and is not a real ditelon. A real ditelon is always filled.
In general, we may say that a real polytope is always filled. In both the above cases, the mapping of the body element to the geometrical figure was straightforward. For polygons and higher polytopes, current abstract theory admits only one kind of filling, which we will consider in the context of other possible fillings.
Any ditelon includes its two bounding vertices. Two ditela may share a common vertex, so mathematically we can order three or more to form a closed circuit, or polygon. Does such a polygon have an inside and an outside? Not necessarily - if the 2-space it inhabits is the surface of say a toroid, it might pass round the toroid as in Figure 11. In this case there can be no filling.
Figure 11 : A star octagon drawn on a toroid.
But usually we obtain the familiar dividing of 2-space into smaller (inside) and larger (outside) regions. We have seen several ways to fill a star polygon. Here they are again in Figure 12, along with a few more.
Figure 12 : Different rules produce different fillings
The 14-sided polygon shown was chosen to higlight the way that different fillings affect the calaculation of the polygon's area. Mathematicians disagree as to how the area of a polygon should be defined, which comes down to disagreeing about how to fill polygons. One common method for finding areas of non-convex polygons divides the figure into triangles (Mark any point. Draw a line connecting each vertex to the point). It then adds or subtracts the area of each triangle, according to which way round the triangle winds (Choose a direction of traversal for the polygon, and mark each side with an arrow as you traverse it. Traverse each triangle in the direction given by the connected arrow). This yields a double-count of area for the region of density 2, the same as the winding rule 10(e). Well, that sounds reasonable - after all it is covered twice. Abstract theory defines the interior, or body, as the set of points which the perimeter contains, 10(c). This yields a single count for any region, regardless of density. Oops! Who's right?
Notice that I have been talking as if there is a single "polygon" that can have different "fillings." It seems reasonable to treat all the objects in Figure 12 as essentially the same polygon rather than different ones. Otherwise, would we not hve great difficulty in defining what a "polygon" is. And having said that, is it reasonable to say that we may fill our polygon in any way we choose, and that different fillings suit different purposes? For example the open cubic box we met earlier has six identifable squares, so why not just say that even though five are filled and one is not, they are all the same polygon?
But consider that open face as a different colouring - suppose we have a blue box with a red lid, and we put the lid back on. Now, if we rotate the cube a quarter-turn so that it lies on its side, the coloured face is no longer in its original position and the mathematical symmetry of the uncoloured cube is brloken. Such colourings play an important part in polyhedral theory, and we recognise different colour arrangements as having different symmetries. For example our coloured cube is no longer a regular polyhedron because it does not have a regular symmetry. This argument only makes sense if we treat the red and blue squares as different polygons.
To be continued...