Order-4 dodecahedral honeycomb

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Order-4 dodecahedral honeycomb

Perspective projection view
within Beltrami-Klein model
Type Hyperbolic regular honeycomb
Schläfli symbol {5,3,4}
{5,31,1}
Coxeter-Dynkin diagram Image:CDW_ring.pngImage:CDW_5.pngImage:CDW_dot.pngImage:CDW_3.pngImage:CDW_dot.pngImage:CDW_4.pngImage:CDW_dot.png
Image:CD_ring.pngImage:CD_5.pngImage:CD_3b.pngImage:CD_downbranch-00.pngImage:CD_3.pngImage:CD_dot.png
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 Order-5 cubic honeycomb
Coxeter group [5,3,4]
[5,31,1]
Properties Regular

The order-4 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.

It can also be constructed from the bifurcating Coxeter group [5,31,1] which can be represented by alternation of two colors of dodecahedral cells.

There is another regular honeycomb in hyperbolic 3-space called the order-5 dodecahedral honeycomb which has 5 dodecahedra per edge.

This honeycomb is also related to the 120-cell which has 120 dodecahedra in 4-dimensional space, 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.

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