This document discusses various polytopes defined by Coxeter groups and their Schläfli symbols. It provides information on the order of the Coxeter groups, the Schläfli symbols that define each polytope, and counts of their facets, vertices, edges, faces and cells. The polytopes range from a line segment and pentagon to the dodecahedron, 120-cell, and a honeycomb of 120-cells.

1. Dodecahedral pentagonal polytopes
n Coxeter group
Petrie polygon
projection
Name
Coxeter diagram
Schläfli symbol
Facets
Elements
Vertices Edges Faces Cells 4-faces
1 [ ]
(order 2)
Line segment
{ }
2 vertices 2
2 [5]
(order 10)
Pentagon
{5}
5 edges 5 5
3 [5,3]
(order 120)
Dodecahedron
{5, 3}
12 pentagons
20 30 12
4 [5,3,3]
(order 14400)
120-cell
{5, 3, 3}
120 dodecahedra
600 1200 720 120

2. 5 [5,3,3,3]
(order ∞)
120-cell honeycomb
{5, 3, 3, 3}
∞ 120-cells
∞ ∞ ∞ ∞ ∞