Infinite Repeating Polyhedra - Partial Honeycombs in 3-Space
Here is a collection of Infinite Repeating Polyhedra, based on regular-polyhedral honeycombs with some of the polyhedra removed, so that only two faces meet at any edge. There are many regular-faced infinite polyhedra out there, but these have the advantage that both the positive and negative space can be subdivided into regular polyhedra. For presentation purposes, I have closed off the last few faces where additional polyhedra would connect, so that my partial models are complete polyhedra, but as each model is extended to infinity, as each component polyhedron is added, the common faces where they join are removed.
 | 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).
Back to the main Polyhedron Page.
Link to this page as http://Polyhedra.Doskey.com/PartialHoneycombs.html