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 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.

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.

The 13 architectonic or catoptric tessellations, shown as uniform cell centers, and catoptric cells, arranged as multiples of the smallest cell on top.
These are four of the 35 cubic space groups