Geometry Page

Symmetry has always been attractive to mathematicians, and the most symmetric of all figures are the regular polyhedra, or Platonic solids. A regular polyhedron is defined as a finite polyhedron composed of a single type of regular polygon such that each element (vertex, edge and face) is surrounded identically. In three dimensions there are exactly five such polyhedra which don't intersect themselves, and four more that do. There are many other interesting such figures, many of which are defined by relaxing one or more of the conditions defining regular polyhedra. For instance, the figure above is composed of only regular triangular faces, but it has three types of edges and three types of vertices. (The three types of vertices are surrounded by 4, 6 and 10 triangles.) Click on the following link for more information on deltahedra.

The following set of images are of some figures I have found which satisfy most or all the criteria defining regular polyhedra except that they are not finite. In other words, it would take an infinite number of polygons to complete such a figure which would then fill all of space with a latticework. Of course an infinite model cannot be completely constructed, but large enough sections can be built to show their geometry and prove their existence. The famous mathematician H.S.M. Coxeter calls these figures "regular skew polyhedrons" while J.R. Gott, III calls them "Pseudopolyhedrons".

Regular polyhedra are often represented with a notation called Schläfli symbols which consist of two numbers between curly braces. The first number is the number of sides on each polygon, and the second is the number of such polygons surrounding each vertex. For example, {4,3} is the cube because each vertex is surrounded by three squares. It's perfectly natural to apply this notation to infinite polyhedra too.

Some images below are in low-res. Simply select the ones with blue borders to see a higher resolution version.

The plastic pieces used to construct the models are called "Polydron" and are the best tools I've seen for exploring geometry. They are hard and smooth, and their hinged edges snap together very easily.

The {3,9} is a particularly pleasing figure. It can easily be described as a set of translations of a simple base figure consisting of four open octahedra connected to the four faces of a tetrahedron. The faces of the tetrahedra are not included, as are the faces of the octahedra which are parallel to the faces of that imaginary tetrahedron.

The {4,6} is very interesting because it is not rigid even when tiled to infinity. Its movement has three degrees of freedom, one for each of the three coordinate axis. It's difficult to construct this model without it collapsing unless these DOF are locked down first by adding any three non-parallel square "braces" into parts of the lattice where they don't belong. A single such "impurity" in the crystal lattice is enough to entirely lock its flexibility in one direction.

Vladimir Bulatov created a wonderful VRML viewer which can be used to view and interactively tile these sort of repeat units in three dimensions. It is currently known to work with the Cosmo 2.1 VRML viewer from SGI and may freeze your browser if you try to use a different one, but it's a really useful feature to be able to change the tiling parameters in order to better understand the structure of these models. You rotate the figure by clicking and dragging on the model, and you change the tiling parameters by clicking on the green pyramids, and change the size of the model with the 3D slider widget. When you have a VRML 2.0 viewer installed, click the following link to see an interactive VRML 2 version of the {4,6} above.

There are many other interesting ways to construct a {4,6}. Here is another way which was suggested by Chaim Goodman-Strauss and John Sullivan:

The image above is really part of a screen shot taken from the VRML version of that {4,6}. There are really several very intersting classes of {4,6} some of which contain uncountably infinite numbers of members. Chaim Goodman-Strauss and John Sullivan have been studying one such class in which all points in the 3D coordinate system with integer coordinates are used as part of each {4,6}. Here is a VRML 2 {4,5} which is not based on a cubic lattice.

I don't have any Polydron hexagons, so the hexagons in models above and below are built from six triangles of the same color. You'll have to imagine that those internal edges don't exist.

The {6,6} pictured above can easily be described as a set of translations of the truncated tetrahedron. Note that the spaces between the truncated tetrahedra are themselves truncated tetrahedra, thereby partitioning space into two identically shaped regions.

I didn't discover the {6,4}, it was described to me by someone that saw it and the {4,6} described in a book by Coxeter. Like the {4,6}, it partitions space into two identical regions.

The three figures above are truly "regular" in that all their edges are identical to the others in each figure. The following figures cannot make that claim and so may be called "semi-regular".

The {5,5} is my most recent and surprising discovery. I'd looked in vien for a {5,4} and had all but given up on finding an infinite polyhedra composed of pentagons. Almost as a lark I tried with five at a vertex and very quickly found the above configuration. I would be surprised if there are any others. I later learned that this figure had been previously discovered and published by J.R. Gott in 1967. The symmetry of this figure is similar to that of the {3,10} below though it is less obvious. It contains a repeating zig-zag motif parallel to two of the three major planes. I only had enough pentagonal Polydron to construct the physical model above, so I also created a larger VRML 1 version of the {5,5} which you can view if your browser only contains a VRML viewer or plug-in, or you have a VRML 2 plug-in, you can use the tiling viewer to see the VRML 2 version of the {5,5}.

This is only one possible configuration of a {3,10}. Notice that as each new layer is added, there are two possible choices as to how to translate the columns. One way to think of this is to notice how each vertex touches one corner of an octahedron from below and one from above. Looking downward at one of the holes in a layer left by, say, a blue octahedron from below, the red octahedra that share the vertices of that blue octahedron, can be arranged rotated sixty degrees in either the clockwise or counter-clockwise directions. The {3,10} shown above was created with all the red octahedra rotated counter-clockwise from the blue octahedra they sit on top of, and with the blue octahedra rotated clockwise from the red octahedra they are on top of.

There are many classes of infinite regular polyhedra with the same layered form as the {3,10} above. Perhaps these classes have an analogous relationship to the general class of infinite regular polyhedra as the prisms and antiprisms have to the finite regular polyhedra. The {4,6} further above is also layered, but it has full cubic symmetry instead of having a "grain" in one dimension. There are many other classes of {4,5}'s and {4,6}'s with the same layered form as the {3,10} above. Linked here is a VRML model of a particularly interesting layered {4,6}. One interesting aspect of this model is that like the first {4,6} above, it is also flexible even when tiled to infinity.

The images above and below are presented in cross-eyed stereo pairs for clarity. If they do not appear side-by-side in your browser, you will need to make your window larger so that they can be seen side-by-side for this effect to work. If you've never viewed stereo image pairs without a viewer, it takes some practice but is well worth the effort. For cross-eyed views like these, you need to cross your eyes until one image from each side exactly overlap in the center. You then try to hold their positions steady while you relax your focus until the image becomes sharp.

The {3,8} shown here is quite beautiful. It forms the same lattice as the bonds between Carbon atoms in diamond crystal. You may also view a VRML 2 version of this {3,8} which I created using the viewer template that Vladimir used for the {5,5} above.

Linked here is another VRML {3,8} which is essentially a cubic packing of snub cubes connected by their square faces leaving only triangles connected 8 at each vertex. The snub cubes are connected in alternating right and left handed versions. The VRML model colors the right handed ones red and the left handed ones green. I found this figure in the book "The Geometrical Foundation of Natural Structure" by Robert Williams although I doubt he realized that the figure constituted an infinite regular polyhedron.

So just how many triangles can surround each vertex and still generate a non self-intersecting infinite polyhedron? I have no idea, but higher numbers are possible. Below is a {3,12} which can be generated from four intersecting copies of the {3,8} above shifted so that their hubs coincide (where "hubs" are where the Carbon atoms sit in the diamond model). I can't build that {3,12} out of Polydron because the dihedral angles become too acute but you can view a VRML 2 version of that {3,12}.

Here is a different {3,12} which can be made from Polydron, from a vertex figure suggested by Don Hatch. It is particularly beautiful being composed of flat snaking paths of coincident triangles that stretch to infinity. The paths come in four different orientations, each shown here in a different color. The result is an impossible seeming figure that looks very much like the intersecting staircases in M.C. Escher's "House of Stairs" lithographs. Below is a view from the inside of the model (with one yellow triangle removed for clarity), seen as if you're walking along a red path.

In this VRML 2 model, paths in each orientation are again rendered the same color but  in alternating shades so that the individual triangles can be distinguished. Here's another version of this same VRML 2 {3,12} but with black edges added instead of alternating colors.

Here's still another VRML 2 {3,12} from another vertex figure by Don Hatch.

Finally, here is a unique and elegant {3,7} shown in stereo. It has the same diamond crystal topology as the {3,8} above, but with icosahedral hubs instead of octahedra:

It's interesting to note that there are two types of hubs in this figure, distinguished here by red and green icosahedra where each type is always connected only to hubs of the other type. The differences between hub types are in the arrangement of their struts. You can call a hub either left or right handed in that if you follow an edge that extends straight out from one of the missing triangles which is the base of one strut, the strut adjacent to the other end of that edge is to the left in the green hubs, and to the right in the red ones. Here is the VRML {3,7}.

Lastly, there exists a beautiful {3,9} which is something like two intersecting {3,7}s. You can visualize it if you imagine a new icosahedron placed in the center of the large voids such as the model above surrounds. Then imagine connecting all 8 neighboring icosahedra connected to that new one. Of course instead of just imagining it you can see the VRML {3,9}.