Hyperbolic small dodecahedral honeycomb

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Small dodecahedral honeycomb
Schläfli symbol	{5,3,4}
Type	regular hyperbolic honeycomb
Cells	dodecahedron {5,3}
Faces	pentagon {5}
Edge figure	square {4}
Vertex figure	octahedron {3,4}
Cells/edge	{5,3}4
Cells/vertex	{5,3}8
Euler characteristic	0
Dual	great cubic honeycomb {4,3,5}
Symmetry group	group [5,3,4]
Properties	Regular

The small dodecahedral honeycomb is one of four regular space-filling tessellation (or honeycombs) in hyperbolic 3-space.

Four dodecahedra exist on each edge, and 8 dodecahedra around each vertex. Its vertices are constructed from 3 orthogonal axes, just like the cubic honeycomb of Euclidean 3-space.

There is another honeycomb in hyperbolic 3-space called the great dodecahedral honeycomb which has 5 dodecahedra per edge instead of 4 here.

This honeycomb is also related to the 120-cell which has 120 dodecahedra on the surface of a 4 dimensional sphere, with 3 dodecahedra on each edge.

The dihedral angle of a dodecahedron is ~116.6°, so it is impossible to fit 4 of them on an edge in Euclidean 3-space. However in hyperbolic space a properly scaled dodecahedron can be scaled so that its dihedral angles are reduced to 90 degrees, and then four fit exactly on every edge.

[edit] See also

• Poincaré sphere Poincaré dodecahedral space
• Seifert-Weber space Seifert-Weber dodecahedral space
• List of regular polytopes

[edit] External links

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• [3]