Order-5 Dodecahedral Honeycomb

The order-5 dodecahedral honeycomb is one of four regular space-filling tessellations (or honeycombs) in hyperbolic 3-space.

Each edge of the honeycomb is surrounded by five dodecahedra exist on each edge, and each vertex is surrounded by twenty dodecahedra.

The dihedral angle of a Euclidean regular dodecahedron is ~116.6°, so no more than three of them can fit around an edge in Euclidean 3-space. In hyperbolic space, however, the dihedral angle is smaller than it is in Euclidean space, and depends on the size of the figure; the smallest possible dihedral angle is 60º, for an ideal hyperbolic regular dodecahedron with infinitely long edges. The dodecahedra in the dodecahedral honeycomb are sized so that all of their dihedral angles are exactly 72º.