Architectonic and catoptric tessellation

Uniform Euclidean 3D tessellations and their duals

In geometry, John Horton Conway defines architectonic and catoptric tessellations as the uniform tessellations 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 which are excluded from these categories.

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