Top Qs
Timeline
Chat
Perspective

Architectonic and catoptric tessellation

Uniform Euclidean 3D tessellations and their duals From Wikipedia, the free encyclopedia

Architectonic and catoptric tessellation
Remove ads
Remove ads

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.

Thumb
The 13 architectonic or catoptric tessellations, shown as uniform cell centers, and catoptric cells, arranged as multiples of the smallest cell on top.
Remove ads

Enumeration

Summarize
Perspective

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.

More information Ref. indices, Symmetry ...
Remove ads

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:

Thumb

Symmetry

Thumb
These are four of the 35 cubic space groups

These four symmetry groups are labeled as:

More information Label, Description ...

References

Loading content...

Further reading

Loading content...
Loading related searches...

Wikiwand - on

Seamless Wikipedia browsing. On steroids.

Remove ads