set as the Architectonic tessellations and the dual honeycombs as the Catoptric tessellations. The individual honeycombs are listed with names given to them by Norman Johnson. (Some of the terms used below are defined in Uniform 4-polytope#Geometric derivations for 46 nonprismatic Wythoffian uniform 4-polytopes) For cross-referencing, they are given with list indices from Andreini (1-22), Williams(1-2,9-19), Johnson (11-19, 21–25, 31–34, 41–49, 51–52, 61–65), and Grünbaum(1-28). Coxeter uses δ4 for a cubic honeycomb, hδ4 for an alternated cubic honeycomb, qδ4 for a quarter cubic honeycomb, with subscripts for other forms based on the ring patterns of the Coxeter diagram. Compact Euclidean uniform tessellations (by their infinite Coxeter group families) The fundamental infinite Coxeter groups for 3-space are: The , [4,3,4], cubic, (8 unique forms plus one alternation) The , [4,31,1], alternated cubic, (11 forms, 3 new) The cyclic group, [(3,3,3,3)] or [3[4]], (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 ×, [4,4,2,∞] prismatic group, (2 new forms) The ×, [6,3,2,∞] prismatic group, (7 unique forms) The ×, [(3,3,3),2,∞] prismatic group, (No new forms) The ××, [∞,2,∞,2,∞] 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. The C̃3, [4,3,4] group (cubic) The regular cubic honeycomb, represented by Schläfli symbol {4,3,4}, offers seven unique derived uniform honeycombs via truncation operations. (One redundant form, the runcinated cubic honeycomb, is included for completeness though identical to the cubic honeycomb.) The reflectional symmetry is the affine Coxeter group [4,3,4]. There are four index 2 subgroups that generate alternations: [1+,4,3,4], [(4,3,4,2+)], [4,3+,4], and [4,3,4]+, with the first two generated repeated forms, and the last two are nonuniform. B̃3, [4,31,1] group The , [4,3] group offers 11 derived forms via truncation operations, four being unique