This page was inspired by a discussion with some friends. We were looking for any regular-faced polyhedra that had at least one vertex where only a single triangle, square and pentagon met.
Below are the solid and frame versions of this 345-hedron and numerous related figures. The first batch are all related to the first figure below. Although we re-discovered this figure, it was previously described by Professor Bonnie Stewart in his book "Adventures Amongst the Toroids" where he labeled it "G_{3}".
You might notice that the bottom of the figure has three intersecting pentagons, and ask "What would happen if I inscribed four of these inside of a dodecahedron?" My first guess was that the top triangle from each of these four figures would meet in the center to form a tetrahedron. I did not notice that the four-coloring of a dodecahedron was chiral, or I would not have even tried this. The result was four intersecting G_{3}s:
Next I tried placing the same four G_{3}s around a central octahedron, touching alternate faces. I knew it would still not be inscribed in a dodecahedron, but I thought it might be a nice model to look at. Please note that the G_{3}s do not quite touch each other at the corners:
This led me to the discovery of a neat conglomeration of polyhedra. I won't go into its derrivation, but you can probably figure it out if you look long enough. Although it really doesn't belong on this page, I don't have a better place for it:
Jim McNeill (who wrote the "hedron" program I used to generate these models) ended up inscribing the four G_{3}s in a dodecahedron before I did (I guess I got too distracted). Make sure you visit his site (Jim's Site) and try out his "hedron" application. He now has his figures posted on his "Stewart G3" page:
Jim's "Stewart G_{3}" Page
Alternatively, if you wanted to remove G_{3}s from a dodecahedron (one or more times) and see what was left, you would get 3455-hedra (two pentagons meeting at the vertex, not one). These models were recommended by John Conway:
If you want to see some cool paper models of G3s extracted from dodecahedra, check out Ulrich Mikloweit's 345 Page
By far the most interesting extraction of G_{3}s from another figure was discovered by B.M. Stewart, in his dissection of the rhombicosidodecahedron. Details of this dissection can be found on my page:
Decomposition of E_{5} - The Rhombicosidodecahedron
I have a few more to share, but I will have to add them here later.
We are still looking for other interesting figures that fall in this category. So if you have any suggestions, please let me know.
The original discussion was started by Melinda Green (please visit his geometry web site), who brought up the topic of searching for polyhedra containing a particular arrangement of faces around a vertex, (which have been dubbed "acrohedra").
For example, the 455 case (one square and two pentagons meeting at one of the vertices of a polyhedron) still has not been solved. If you have a solution for this case, please let me know. Although I must warn you that there has been speculation that no such solution exists. I have constructed a few "near misses" out of polygon construction toys, but they always required a little bit of "pressure" to get them to close properly (the dihedral angles weren't quite correct).
Jim McNeill now has a very good page of acrohedra that have been discovered. Please check it out:
Jim's "Acrohedra" Page