Uniform 5-polytope
These construction operations are represented by the permutations of rings of the Coxeter diagrams.
All except the grand antiprism prism are based on Wythoff constructions, reflection symmetry generated with Coxeter groups.
Each reflective uniform 5-polytope can be constructed in one or more reflective point group in 5 dimensions by a Wythoff construction, represented by rings around permutations of nodes in a Coxeter diagram.
Mirror hyperplanes can be grouped, as seen by colored nodes, separated by even-branches.
Uniform polytopes in these group with symmetric rings contain this extended symmetry.
If all the nodes of a given color are ringed (active), an alternation operation can generate a new 5-polytope with chiral symmetry, shown as "empty" circled nodes", but the geometry is not generally adjustable to create uniform solutions.There are 5 finite categorical uniform prismatic families of polytopes based on the nonprismatic uniform 4-polytopes.
That brings the tally to: 19+31+8+45+1=104 In addition there are: There are 19 forms based on all permutations of the Coxeter diagrams with one or more rings.
(16+4-1 cases) They are named by Norman Johnson from the Wythoff construction operations upon regular 5-simplex (hexateron).
7 of the 19 figures, with symmetrically ringed Coxeter diagrams have doubled symmetry, order 1440.
The coordinates of uniform 5-polytopes with 5-simplex symmetry can be generated as permutations of simple integers in 6-space, all in hyperplanes with normal vector (1,1,1,1,1,1).
This family has 25−1=31 Wythoffian uniform polytopes generated by marking one or more nodes of the Coxeter diagram.
Also added are 8 uniform polytopes generated as alternations with half the symmetry, which form a complete duplicate of the D5 family as ... = ..... (There are more alternations that are not listed because they produce only repetitions, as ... = .... and ... = ....
These would give a complete duplication of the uniform 5-polytopes numbered 20 through 34 with symmetry broken in half.)
The 5-cube family of 5-polytopes are given by the convex hulls of the base points listed in the following table, with all permutations of coordinates and sign taken.
This family has 23 Wythoffian uniform polytopes, from 3×8-1 permutations of the D5 Coxeter diagram with one or more rings.
In the 15 repeats, both of the nodes terminating the length-1 branches are ringed, so the two kinds of element are identical and the symmetry doubles: the relations are ... = .... and ... = ..., creating a complete duplication of the uniform 5-polytopes 20 through 34 above.
The last three snubs can be realised with equal-length edges, but turn out nonuniform anyway because some of their 4-faces are not uniform 4-polytopes.
Uniform duoprism prisms, {p}×{q}×{ }, form an infinite class for all integers p,q>2.
The full set of uniform polytopes generated are based on the unique permutations of ringed nodes.
The prismatic forms and bifurcating graphs can use the same truncation indexing notation, but require an explicit numbering system on the nodes for clarity.
[11][12] There are three regular honeycombs of Euclidean 4-space: Other families that generate uniform honeycombs: Non-Wythoffian uniform tessellations in 4-space also exist by elongation (inserting layers), and gyration (rotating layers) from these reflective forms.
= [5,3,3,5]: There are 5 regular compact convex hyperbolic honeycombs in H4 space:[13] There are also 4 regular compact hyperbolic star-honeycombs in H4 space: There are 9 paracompact hyperbolic Coxeter groups of rank 5, each generating uniform honeycombs in 4-space as permutations of rings of the Coxeter diagrams.
Paracompact groups generate honeycombs with infinite facets or vertex figures.
