Compact Euclidean Uniform Tessellations (by Their Infinite Coxeter Group Families)
The fundamental infinite Coxeter groups for 3-space are:
- The, cubic, (8 unique forms plus one alternation)
- The, alternated cubic, (11 forms, 3 new)
- The cyclic group, or )], (5 forms, one new)
There is a correspondence between all three families. Removing one mirror from produces, and removing one mirror from produces . This allows multiple constructions of the same honeycombs. If cells are colored based on unique positions within each Wythoff construction, these different symmetries can be shown.
In addition there are 5 special honeycombs which don't have pure reflectional symmetry and are constructed from reflectional forms with elongation and gyration operations.
The total unique honeycombs above are 18.
The prismatic stacks from infinite Coxeter groups for 3-space are:
- The x, x prismatic group, (2 new forms)
- The x, x prismatic group, (7 unique forms)
- The x, (3 3 3)x prismatic group, (No new forms)
- The xx, xx prismatic group, (These all become a cubic honeycomb)
In addition there is one special elongated form of the triangular prismatic honeycomb.
The total unique prismatic honeycombs above (excluding the cubic counted previously) are 10.
Combining these counts, 18 and 10 gives us the total 28 uniform honeycombs.
Read more about this topic: Convex Uniform Honeycomb
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