Message #2257

From: schuma <>
Subject: Regular abstract polytopes based on {5,3,4} and {4,3,5}
Date: Fri, 08 Jun 2012 05:48:40 -0000


The regular abstract polytopes based on hyperbolic tessellations {5,3,4} and {4,3,5} have been mentioned by Andrey several times here. Recently I read more about them and found Gruenbaum talked about a polytope formed by 32 hemidodecahedra, which was related to {5,3,4}. It should be this one:

According to this page, it has 32 cells, each of which is a hemidodecahedron. It has 96 faces, 120 edges and 40 vertices. The vertex figure is an octahedron (note: not hemi-octahedron). Compared with the 11-cell and the 57-cell, this 32-cell received little attention.

It has a dual, which is based on {4,3,5}:

The 40 faces are cubes (not hemi-cubes). The vertex figure is a hemi-icosahedron.

The vertex coordinates of {4,3,5} and {5,3,4} have been computed analytically and can be found in this paper [Garner, Coordinates for Vertices of Regular Honeycombs in Hyperbolic Space,, Table 1]. This is of course a good news for implementation. If any one wants to see the paper but has no access to Jstor please email me. The coordinates are in a Minkowskian space. I need to learn more hyperbolic geometry to understand the model.

According to Colbourn and Weiss [A CENSUS OF REGULAR 3-POLYSTROMA ARISING FROM HONEYCOMBS,], there are more abstract polytopes based on {5,3,4} and {4,3,5}. But they cannot be found in [] because this atlas contains information of "small" polytopes with up to 2000 symmetries. Fortunately, the 32-hemidodecahedral-cell and its dual, 40-cubic-cell have 1920 symmetries, which is just below the boundary. Something like 120-cell, and 57-cell etc are not there because they are too large. But these two things can keep me excited for a while.