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In the geometry of hyperbolic 3-space, the order-4 dodecahedral honeycomb is one of four compact regular space-filling tessellations (or honeycombs). With Schläfli symbol {5,3,4}, it has four dodecahedra around each edge, and 8 dodecahedra around each vertex in an octahedral arrangement. Its vertices are constructed from 3 orthogonal axes. Its dual is the order-5 cubic honeycomb.
Contents
- Description
- Symmetry
- Images
- Related polytopes and honeycombs
- Rectified order 4 dodecahedral honeycomb
- Related honeycombs
- Truncated order 4 dodecahedral honeycomb
- Bitruncated order 4 dodecahedral honeycomb
- Cantellated order 4 dodecahedral honeycomb
- Cantitruncated order 4 dodecahedral honeycomb
- Runcitruncated order 4 dodecahedral honeycomb
- References
A geometric honeycomb is a space-filling of polyhedral or higher-dimensional cells, so that there are no gaps. It is an example of the more general mathematical tiling or tessellation in any number of dimensions.
Honeycombs are usually constructed in ordinary Euclidean ("flat") space, like the convex uniform honeycombs. They may also be constructed in non-Euclidean spaces, such as hyperbolic uniform honeycombs. Any finite uniform polytope can be projected to its circumsphere to form a uniform honeycomb in spherical space.
Description
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.
Symmetry
It a half symmetry construction, {5,31,1}, with two types (colors) of hexagonal tilings in the Wythoff construction. ↔ .
Images
Beltrami-Klein model
Related polytopes and honeycombs
There are four regular compact honeycombs in 3D hyperbolic space:
There are fifteen uniform honeycombs in the [5,3,4] Coxeter group family, including this regular form.
There are eleven uniform honeycombs in the bifurcating [5,31,1] Coxeter group family, including this honeycomb in its alternated form. This construction can be represented by alternation (checkerboard) with two colors of dodecahedral cells.
This honeycomb is also related to the 16-cell, cubic honeycomb, and order-4 hexagonal tiling honeycomb all which have octahedral vertex figures:
This honeycomb is a part of a sequence of polychora and honeycombs with dodecahedral cells:
Rectified order-4 dodecahedral honeycomb
The rectified order-4 dodecahedral honeycomb, , has alternating octahedron and icosidodecahedron cells, with a cube vertex figure.
Related honeycombs
There are four rectified compact regular honeycombs:
Truncated order-4 dodecahedral honeycomb
The truncated order-4 dodecahedral honeycomb, , has octahedron and truncated dodecahedron cells, with a cube vertex figure.
It can be seen as analogous to the 2D hyperbolic truncated order-4 pentagonal tiling, t{5,4} with truncated pentagon and square faces:
Bitruncated order-4 dodecahedral honeycomb
The bitruncated order-4 dodecahedral honeycomb, or bitruncated order-5 cubic honeycomb, , has truncated octahedron and truncated icosahedron cells, with a tetrahedron vertex figure.
Cantellated order-4 dodecahedral honeycomb
The cantellated order-4 dodecahedral honeycomb,, has rhombicosidodecahedron and cuboctahedron, and cube cells, with a triangular prism vertex figure.
Cantitruncated order-4 dodecahedral honeycomb
The cantitruncated order-4 dodecahedral honeycomb, is a uniform honeycomb constructed with a coxeter diagram, and mirrored sphenoid vertex figure.
Runcitruncated order-4 dodecahedral honeycomb
The runcititruncated order-4 dodecahedral honeycomb, is a uniform honeycomb constructed with a coxeter diagram, and a quadrilateral pyramid vertex figure.