Uniform 6-polytope

These construction operations are represented by the permutations of rings of the Coxeter-Dynkin diagrams.

Each combination of at least one ring on every connected group of nodes in the diagram produces a uniform 6-polytope.

The simplest uniform polypeta are regular polytopes: the 6-simplex {3,3,3,3,3}, the 6-cube (hexeract) {4,3,3,3,3}, and the 6-orthoplex (hexacross) {3,3,3,3,4}.

Uniform 6-polytopes with reflective symmetry can be generated by these four Coxeter groups, represented by permutations of rings of the Coxeter-Dynkin diagrams.

There are four fundamental reflective symmetry groups which generate 153 unique uniform 6-polytopes.

Each combination of at least one ring on every connected group produces a uniform prismatic 6-polytope.

These fundamental families generate 153 nonprismatic convex uniform polypeta.

In addition, there are infinitely many uniform 6-polytope based on: There are 32+4−1=35 forms, derived by marking one or more nodes of the Coxeter-Dynkin diagram.

They are named by Norman Johnson from the Wythoff construction operations upon regular 6-simplex (heptapeton).

The coordinates of uniform 6-polytopes with 6-simplex symmetry can be generated as permutations of simple integers in 7-space, all in hyperplanes with normal vector (1,1,1,1,1,1,1).

There are 63 forms based on all permutations of the Coxeter-Dynkin diagrams with one or more rings.

They are named by Norman Johnson from the Wythoff construction operations upon the regular 6-cube and 6-orthoplex.

This family has 3×16−1=47 Wythoffian uniform polytopes, generated by marking one or more nodes of the D6 Coxeter-Dynkin diagram.

Uniform triaprisms, {p}×{q}×{r}, form an infinite class for all integers p,q,r>2.

The full set of uniform polytopes generated are based on the unique permutations of ringed nodes.

Uniform 6-polytopes are named in relation to the regular polytopes in each family.

The prismatic forms and bifurcating graphs can use the same truncation indexing notation, but require an explicit numbering system on the nodes for clarity.