# Architectonic and catoptric tessellation

In geometry, John Horton Conway defines architectonic and catoptric tessellations as the uniform tessellations (or honeycombs) of Euclidean 3-space with prime space groups 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 the corresponding catoptric tessellation, and vice versa. The cubille is the only Platonic (regular) tessellation of 3-space, and is self-dual. There are other uniform honeycombs constructed as gyrations or prismatic stacks (and their duals) which are excluded from these categories.

## Enumeration

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.
| Ref. indices | Symmetry | Architectonic tessellation | Architectonic tessellation | Architectonic tessellation | Catoptric tessellation | Catoptric tessellation | Catoptric tessellation |
| Ref. indices | Symmetry | Name Coxeter diagram Image | Vertex figure Image | Cells | Name | Cell | Vertex figures |
| J$_{11,15}$ A$_1$ W$_1$ G$_{22}$ δ$_4$ | nc [4,3,4] | Cubille (Cubic honeycomb) | Octahedron, | | Cubille | Cube, | |
| J$_{12,32}$ A$_{15}$ W$_{14}$ G$_7$ t$_1$δ$_4$ | nc [4,3,4] | Cuboctahedrille (Rectified cubic honeycomb) | Cuboid, | | Oblate octahedrille | Isosceles square bipyramid | , |
| J$_{13}$ A$_{14}$ W$_{15}$ G$_8$ t$_{0,1}$δ$_4$ | nc [4,3,4] | Truncated cubille (Truncated cubic honeycomb) | Isosceles square pyramid | | Pyramidille | Isosceles square pyramid | , |
| J$_{14}$ A$_{17}$ W$_{12}$ G$_9$ t$_{0,2}$δ$_4$ | nc [4,3,4] | 2-RCO-trille (Cantellated cubic honeycomb) | Wedge | | Quarter oblate octahedrille | irr. Triangular bipyramid | , , |
| J$_{16}$ A$_3$ W$_2$ G$_{28}$ t$_{1,2}$δ$_4$ | bc [[4,3,4]] | Truncated octahedrille (Bitruncated cubic honeycomb) | Tetragonal disphenoid | | Oblate tetrahedrille | Tetragonal disphenoid | |
| J$_{17}$ A$_{18}$ W$_{13}$ G$_{25}$ t$_{0,1,2}$δ$_4$ | nc [4,3,4] | n-tCO-trille (Cantitruncated cubic honeycomb) | Mirrored sphenoid | | Triangular pyramidille | Mirrored sphenoid | , , |
| J$_{18}$ A$_{19}$ W$_{19}$ G$_{20}$ t$_{0,1,3}$δ$_4$ | nc [4,3,4] | 1-RCO-trille (Runcitruncated cubic honeycomb) | Trapezoidal pyramid | | Square quarter pyramidille | Irr. pyramid | , , , |
| J$_{19}$ A$_{22}$ W$_{18}$ G$_{27}$ t$_{0,1,2,3}$δ$_4$ | bc [[4,3,4]] | b-tCO-trille (Omnitruncated cubic honeycomb) | Phyllic disphenoid | | Eighth pyramidille | Phyllic disphenoid | , |
| J$_{21,31,51}$ A$_2$ W$_9$ G$_1$ hδ$_4$ | fc [4,3$^{1,1}$] | Tetroctahedrille (Tetrahedral-octahedral honeycomb) or | Cuboctahedron, | | Dodecahedrille or | Rhombic dodecahedron, | , |
| J$_{22,34}$ A$_{21}$ W$_{17}$ G$_{10}$ h$_2$δ$_4$ | fc [4,3$^{1,1}$] | truncated tetraoctahedrille (Truncated tetrahedral-octahedral honeycomb) or | Rectangular pyramid | | Half oblate octahedrille or | rhombic pyramid | , , |
| J$_{23}$ A$_{16}$ W$_{11}$ G$_5$ h$_3$δ$_4$ | fc [4,3$^{1,1}$] | 3-RCO-trille (Cantellated tetrahedral-octahedral honeycomb) or | Truncated triangular pyramid | | Quarter cubille | irr. triangular bipyramid | |
| J$_{24}$ A$_{20}$ W$_{16}$ G$_{21}$ h$_{2,3}$δ$_4$ | fc [4,3$^{1,1}$] | f-tCO-trille (Cantitruncated tetrahedral-octahedral honeycomb) or | Mirrored sphenoid | | Half pyramidille | Mirrored sphenoid | |
| J$_{25,33}$ A$_{13}$ W$_{10}$ G$_6$ qδ$_4$ | d [[3$^{[4]}$]] | Truncated tetrahedrille (Cyclotruncated tetrahedral-octahedral honeycomb) or | Isosceles triangular prism | | Oblate cubille | Trigonal trapezohedron | |

## Vertex Figures

The vertex figures of all architectonic honeycombs, and the dual cells of all catoptric honeycombs are shown below, at the same scale and the same orientation:

## 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 | (221) Im3m | I43 | [[4,3,4]] | 8°:2 |
| nc | normal cubic symmetry | (229) Pm3m | P43 | [4,3,4] | 4$^−$:2 |
| fc | half-cubic symmetry | (225) Fm3m | F43 | [4,3$^{1,1}$] = [4,3,4,1$^+$] | 2$^−$:2 |
| d | diamond symmetry or extended quarter-cubic symmetry | (227) Fd3m | F$_d$4$_n$3 | [[3$^{[4]}$]] = [[1$^+$,4,3,4,1$^+$]] | 2$^+$:2 |