Labeled Isogonal Polyhedra

NOTE:  This web site has been set up as a mirror of the one at the University of Washington because that one's continuing future existence is somewhat uncertain. Therefore it would be best if you updated your bookmarks to this new site. But other than some explanatory words on this issue, the two sites should be exactly the same.

Branko Grünbaum and I began this work at the University of Washington many years ago and now I am continuing it. Our goal is to create a comprehensive textual and visual catalogue of all of the labeled isogonal polyhedra that are known. But what is an isogonal polyhedron? That requires some explanation.

An isogonal polyhedron is a polyhedron that essentially "looks the same" from every vertex point. Much research has been done over the centuries on these but many of them were first missed by researchers and it has taken many attempts by multiple people to find those known so far. Most of the previous research has focused on finite polyhedra, that is, those that can be completely confined within a fixed, bounded sphere. However we are primarily interested in extending the known list of those that are infinite in size, those that continue on in one, two, or three dimensions without bound. As if this were not enough work by itself (!), we also want to list all of the possible labelings of these same isogonal polyhedra. This requires even more explanation.

Isogonal polyhedra are also called vertex-transitive polyhedra, and this definition, while longer, gets more directly to what makes them separate from general polyhedra. Consider a "point," or vertex of a polyhedron. It has multiple polygons attached to it. We define a vertex star to be the set of polygons arranged around this central vertex where the dihedral angles between each polygon are fixed. In an isogonal polyhedron every vertex star in the polyhedron is the same. That is, they are all congruent. But we also impose an additional requirement, that of vertex-transitivity. To explain this, consider two identical copies of a polyhedron. To be vertex-transitive it must be possible to position any vertex of the second polyhedron exactly in the same location as any vertex of the first polyhedron and then be able to pivot (and/or reflect, if necessary) the rest of the second polyhedron around that fixed vertex so that both polyhedra coincide perfectly. That is, any vertex can be transposed on top of any other without it being observable that the transposition has occurred. This is what is meant by vertex-transitive. A cube is an example of an isogonal polyhedron:  as seen from any corner, it looks exactly the same.

It may seem like requiring each vertex star to be congruent would automatically produce vertex-transitivity, but this is not correct. Consider the rhombicuboctahedron, which is an isogonal polyhedron. To construct a rhombicuboctahedron consider a belt of eight squares arranged like an octagon. From each of four of the squares, alternating around the belt, add a new square sloping up to the space above the middle of the belt to meet at another, fifth square, forming a "roof." The four empty spaces between the squares will be triangular, so fill them in with triangles. Then do the same on the bottom, using the same four squares in the belt as starting points. This creates the rhombicuboctahedron. Every vertex star consists of three squares and a triangle, all using the same dihedral angles, so all of the vertex stars are congruent. No matter which vertex you look at, the whole polyhedron looks exactly the same. But now create the pseudorhombicuboctahedron. Starting with the rhombicuboctahedron, rotate the five squares and four triangles forming the top "roof" by one square along the belt so that the squares forming the top roof rise up from different squares than those dropping down from the belt to the "floor" below. Now the polyhedron has an asymmetry. That is, when looking at certain different vertices the polyhedron will no longer look exactly the same. Yet all of the vertex stars are still congruent since each one still has three squares and a triangle with the same angular relationships between them. Thus the pseudorhombicuboctahedron is not an isogonal polyhedron, even though all of its vertex stars are congruent.

When working with the plain vertex stars that form a cube it is not possible to do anything but create an isogonal polyhedron because the cube corner is perfectly symmetric. But if the vertex star has an asymmetry, like the vertex star used in the rhombicuboctahedron, it is possible to create a polyhedron that is not isogonal. So how can it be known whether an arbitrary polyhedron is isogonal or not? We can look at every vertex, but if there are an infinite number of vertices this plan could take a very long time!

The solution is to first label the edges emanating from the central vertex in the vertex star. For example, there are three in the cube corner, so we label them a, b, and c. In the rhombicuboctahedron, with three squares and a triangle, there will be four edges:  a, b, c, and d. Once we have established the pattern of which edge in the first vertex star has which label, we can now label every edge in every vertex star in the polyhedron according to the exact same pattern. Now every edge leaving any vertex star must also enter a second vertex star and be one of its edges. And that edge in that second vertex star will have its own label. By noting which edge connects to which other edge we create a connection rule. For example, if the 'a' edge in one vertex star connects to the 'a' edge in its adjacent vertex star then we know that every 'a' edge in every vertex star must connect to an 'a' edge. Because if this were not true then the polyhedron would not look the same from every vertex; but we know that it must because it is vertex-transitive.

This is a very powerful idea. Because now, once we have created a labeled vertex star and established a connection rule, we can start with a single vertex star and then build outward, using new copies of the same vertex star connected according to our connection rule. If we do this, and if what results forms a polyhedron, then we will know absolutely that it is an isogonal polyhedron. This answers our original question. Moreover, not only does this idea tell us how to build an isogonal polyhedron but it also gives us a way to individually name them:  any two isogonal polyhedra that have different labeling rules must be different, and any two that are built the same way must look the same. And as will be discussed later, this idea permits yet a third possibility:  once we have a labeled vertex star we can then make a list of all possible connection rules and then check each one to see if it creates a valid polyhedron. If we follow this technique we can be be very confident that we have not missed any possible isogonal polyhedra that could be built with that vertex star, which has been a problem with earlier research.

We now provide a brief introduction to our labeling system using the cube as an example. As mentioned above, each edge is given a letter but also (in most cases) some additional symbols. The list of labels for all of the edges of the vertex star makes up the vertex symbol. The adjacent edge labels allowed for each original edge label (the connection rule) are also described using the same set of letters and symbols and listed in the adjacency symbol. Together, the vertex symbol and adjacency symbol make up the polyhedron's incidence symbol. As an example, the vertex symbol for a cube corner might be  a+b+c+. This could be paired with an adjacency symbol of  a+c+b+, meaning that the 'a' edges must be paired together while 'b' edges must always be paired with 'c' edges, and together these create the incidence symbol [a+b+c+;  a+c+b+]. (In this case, the '+' signs mean that the vertex stars are used as originally given, without being reflected or otherwise altered.) Of course it is obvious that eight cube corners fit together to make a cube but it may seem surprising that they can still do this when the three edges of the corners are labeled in such a way as to make them distinct while still being required to pair up isogonally. Yet it is indeed possible. In fact, it can be done in eight different ways!

A cube is an isogonal polyhedron but, in particular, it is a finite, convex, isogonal polyhedron. All of the five Platonic solids are finite, convex, isogonal polyhedra. But isogonal polyhedra do not have to be convex, or even finite. While finite polyhedra are well-known, especially the convex ones, polyhedra of infinite size have been little studied. We are trying to find as many of them as we can.

Infinite isogonal polyhedra have been called sponges, infinite skew polyhedra, and pseudopolyhedra, along with various other names. (In two dimensions only, they are usually called tilings.) There is some debate over whether 'sponge' can refer to any non-finite isogonal polyhedron or only to those that are infinite in all three dimensions. In fact, there is debate over whether non-finite isogonal polyhedra are true polyhedra at all, but must be referred to by some separate term. We do not choose to enter into this argument and accept all of these as 'isogonal polyhedra', while referring to those that are only infinite in one dimension as rods, those in only two dimensions as slabs (or tilings), and those that are infinite in all three dimensions as sponges. But these are terms intended merely for easier identification, not terms to be used as implying precise definitions.

As might be expected, allowing infinite polyhedra greatly increases the number of possible sponges that we might turn up. Mostly we are first limiting ourselves to acoptic polyhedra. Acoptic means that no vertices, edges, or faces intersect any other, other than when they meet normally within a vertex star. We make this limitation for three reasons. First, this is more in line with the "classical" notion of what constitutes a polyhedron. Second, it is very difficult to visualize non-acoptic sponges when they are infinitely twisting around, into, and on top of themselves. And finally, there are simply enough acoptic sponges to look at to keep us very busy without worrying about the non-acoptic ones! But, later, we hope to include some of the more accessible ones.

We also mention some details about notation. Vertex stars are usually notated according to the number of edges in the regular polygons making up the star, listed in their cyclical order around the central vertex and separated by periods. For example, the cube has a (4.4.4) vertex star since it consists of three squares. If the polygons are all congruent (as in the cube) then this is often abbreviated using the Schläfli symbol, as in {4,3}  (4-gons, or squares, using 3 of them). Other researchers sometimes use 43 for this cube corner case, though we prefer the first notation.

Not surprisingly, it can be quite tedious to manually search for all possible polyhedra, and even more so to search for all possible unique isogonal labelings of a given polyhedron. Therefore, we have created a computer program to help with this search. One advantage of using the computer is that, for each valid polyhedron and incidence symbol it finds, it can construct 3-dimensional pictures of the polyhedra which can be very helpful in visualizing them. For these pictures we use VRML97 (Virtual Reality Modeling Language, 1997) as the modeling language to display them. VRML97 is the current name for what used to be called VRML 2.0. The new standard now for displaying 3-D objects on the Web is X3D, but X3D has been designed to also read VRML97 files so these images should still be viewable for some time. In any event, in order to view the images, you will need a VRML97- or X3D-compatible standalone program or Web browser, or a plug-in to your regular browser. The Web3D organization maintains a Web page listing such software. To use the standalone programs you will first need to download the VRML files from here to your own computer (they are just plain text files). Here is a simple octahedron for you to use to try out your display software.

More details on all of this are provided in the pages below. I hope you enjoy your visit to these virtual mathematical worlds!

Steven Gillispie

A complete explanation of incidence symbols

More details on the computer program and the VRML images

Results of some vertex star searches

So far we have analyzed a number of vertex stars, which are listed below. Each page shows the vertex and adjacency symbols, along with the VRML images, for each of the polyhedra found. We distinguish between Platonic polyhedra, where all of the regular polygons in the vertex star are the same, and uniform polyhedra, where the vertex star contains mixed polygons.

Platonic polyhedra

Uniform polyhedra

A special tribute...

I continue to search for interesting specimens and their resultant shapes, so feel free to suggest your favorite vertex star of interest. Any comments or corrections you would like to e-mail me would be appreciated.

Last updated: July 5, 2019