Talk:Order-4 dodecahedral honeycomb

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The Poincare Dodecahedron is a surchoron (cell) of the {5,3,3}, a dodecahedron where the cells are linked three at an edge. It can be equally represented by a set of five tetrahedra surrounding an edge, or four tetrahedra around a core tetrahedron.

The Siefert-Weber is a surchoron (cell) of {5,3,5}, having five dodecahedra linked at an edge.

The cells of {5,3,4} can not be used singularly to represent a finite space of the tyoe of Poincare-Dodecahedron, or the Siefert-Weer cell, because the dodecahedron has thirty edges, and thirty is not a multiple of four. However, one can take these in pairs to represent such a space, or convert the dodecahedron into a rhombic tricontahedron, in much the same way that alternate cells of the cubic tiling {4,3,4} can be converted into rhombic dodecahedra to give the dual of {3,4,A}. One can do the same with the {3,3,4}, producing rhombic hexahedra (cubes), to give {4,3,3}.

--Wendy.krieger 08:41, 6 October 2007 (UTC)

[edit] References

Coxeter's 'regular polytopes' does not discuss either hyperbolic polytopes nor "modular" figures (finite maps). The article in 'twelve essays' (regular hyperbolic polytopes) discusses the regular tilings (only), with finite content symmetry cells. One could turn to Coxeter + Mosser "Generators and Relations for Discrete Groups [Second Edition]" deals with representing finite groups by 2d tilings, including hyperbolic ones. --Wendy.krieger 09:41, 20 October 2007 (UTC)