What makes a polyhedron interesting or attractive? This is obviously an open-ended question with many possible answers. One way to approach the problem of designing interesting polyhedra is to note our natural affinity for symmetry and regularity in art and design. We may then describe a family of polyhedra by choosing some set of symmetry and regularity properties we think might be interesting and seeing what polyhedra have those properties. The properties we choose determine who gets invited to the party. The trick is to find a set of properties that open the door just wide enough to let in a bunch of cool, interesting polyhedra, without opening it so wide that all the boring, annoying polyhedra show up too.
The problem with working this way is that most the obvious choices of properties are already spoken for. Lots of people have worked on this problem before, and they've figured out some well-known families of polyhedra. Here's a table summarizing some of those families. Click on the names for more information.
Because all of these families are so well-known, we have to start looking further afield to come up with interesting new polyhedra. We knew that we liked regular faces and convexity, but as you can see above that combination is captured by the Johnson solids. So we decided to loosen up on "all faces regular", and asked only that many faces be regular. Obviously, this property is a bit fuzzy, so there's no right way to enumerate all such polyhedra. Instead, we devise an explicit notation and construction technique that yields polyhedra in this family. Because our construction builds polyhedra by starting with a symmetry group, we call them Symmetrohedra:
If you're interested in finding out more about how Symmetrohedra are constructed and how to interpret the symbol that defines one, you can read the paper that George Hart and I wrote on the subject. It appeared in the 2001 Bridges conference. If, on the other hand, you're simply interested in looking at a gallery of interesting and often novel polyhedra, I've included one here. All the images below were produced using a modified version of a polyhedron viewer written by Doug Zongker. I added code to produce semi-transparent SVG renderings that are then turned into images using Batik.
Platonic and Archimedean SolidsNote that Symmetrohedra can't be used to represent all 18 Platonic and Archimedean solids. The way we defined them, the resulting polyhedra will have all the symmetries of one of the polyhedral groups generated by reflections, and so we can't represent the two snub Archimedeans (though we could with a suitable extension to our notation). The other 16 are all possible.
Interesting and New Polyhedra
The following list of polyhedra is not intended to be definitive or complete in some way. It's merely a sampling of interesting polyhedra we encountered that can be expressed as Symmetrohedra. Many of these polyhedra have never been documented before to our knowledge.
I have dozens more examples on file for this section. I'll add them in as time permits.