04=P=E4.Stel | E4 = Rhombicuboctahedra | |
04=3+2=B4+P4.Stel | B4+P4 = Cuboctahedra + Cubes | |
05=P=3=K3.Stel | K3 = Truncated Octahedra | |
06=P=K4.Stel | K4 = Truncated Cuboctahedra | |
06=3+2=K3+P4.Stel | K3+P4 = Truncated Octahedra + Cubes | |
07=P+4=T4+P8.Stel | T4+P8 = Truncated Cubes + Oct Prisms | |
07=P+3=T4+E4.Stel | T4+E4 = Truncated Cubes + Rhombicuboctahedra | |
07=4+2=P8+P4.Stel | P8+P4 = Oct Prisms + Cubes | |
07=3+2=E4+P4.Stel | E4+P4 = Rhombicuboctahedra + Cubes | |
08=P+4=3+2=K4+P8.Stel | K4+P8 = Truncated Cuboctahedra + Oct Prisms Connected by squares | |
08=P+3=K4+K4.Stel | K4+K4 = Truncated Cuboctahedra + Truncated Cuboctahedra | |
08=P+2=4+3=P8+K4.Stel | P8+K4 = Oct Prisms + Truncated Cuboctahedra Connected by Octagons | |
08=4+2=P8+P8.Stel | P8+P8 = Oct Prisms + Oct Prisms | |
12=P+4=E4+P4.Stel | E4+P4 = Rhombicuboctahedra + Cubes (duplicates 07-3+2) | |
13=P+4=4+2=B4+K3.Stel | B4+K3 = Cuboctahedra + Truncated Octahedra | |
13=P+3=K3+T3_B4_T3.Stel | K3+T3_B4_T3 = Truncated Octahedra + Truncated Tetrahedra + Cuboctahedra + Truncated Tetrahedra |
Below is a table of the Uniform Honeycombs, and the different components that make it up. I took a single polyhedron with octahedral symmetry from each honeycomb, and looked how it was connected to the next of its kind via its 4-fold axes, 3-fold axes and 2-fold axes. By taking a subset of these components, P, 4, 3 and 2, you can check to see if it forms an Infinite Repeating Polyhedron (IRP) with exactly two faces meeting at each edge.
Unif # | Poly | 4-Fold | 3-Fold | 2-Fold | Partials | Compliments |
1 | P4 | square | point | edge | None | None |
2 | B4 | gyr-square | S3 | point | None | None |
3 | T4 | octagon | S3 | edge | None | None |
4 | E4 | square | B4 | P4 | P, 3+2 | P+(3+2) |
5 | K3 | gyr-square | K3 | orth-square | P, 3 | P+3 |
6 | K4 | octagon | K3 | P4 | P, 3+2 | P+(3+2) |
7 | T4 | P8 | E4 | edge-P4 | P+4, P+3, 4+2, 3+2 | (P+4)+(3+2), (P+3)+(4+2) |
8 | K4 | P8 | K4 | diag-P8 | P+4, P+3, P+2, 4+3, 4+2, 3+2 | (P+4)+(3+2) self, (P+3)+(4+2) (P+2)+(4+3) self |
11 | S3 | point | Y3/Y3 | edge-S3 | None | None |
12 | E4 | P4 | Y3/d-P4/Y3 | E4 | P+4 | None |
13 | K3 | B4 | T3/B4/T3 | edge-K3 | P+3, 4+2 | (P+3)+(4+2) |
Thanks to Robert Webb for his very cool "Stella" program. I have used it extensively to generate VRML files for my site. It is a great tool for rapid exploration of augmentations and excavations of polyhedra (among other things).