Architectonic and catoptric tessellation
In geometry, John Horton Conway defines architectonic and catoptric tessellations as the uniform tessellations of Euclidean 3-space and their duals, as three-dimensional analogue of the Platonic, Archimedean, and Catalan tiling of the plane. The singular vertex figure of an architectonic tessellation is the dual of the cell of catoptric tessellation. The cubille is the only Platonic tessellation of 3-space, and is self-dual. There are other uniform honeycombs constructed as prismatic stacks which are excluded from these categories.
The pairs of architectonic and catoptric tessellations are listed below with their symmetry group. These tessellations only represent four symmetry space groups, and also all within the cubic crystal system. Many of these tessellations can be defined in multiple symmetry groups, so in each case the highest symmetry is expressed.
Symmetry
These four symmetry groups are labeled as:| Label | Description | space group Intl symbol | Geometric notation | Coxeter notation | Fibrifold notation |
| bc | bicubic symmetry or extended cubic symmetry | Imm | I43 | 8°:2 | |
| nc | normal cubic symmetry | Pmm | P43 | 4−:2 | |
| fc | half-cubic symmetry | Fmm | F43 | = | 2−:2 |
| d | diamond symmetry or extended quarter-cubic symmetry | Fdm | Fd4n3 | 3 = 1+,4,3,4,1+ | 2+:2 |