| 词条 | Order-5 dodecahedral honeycomb | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 释义 |
The order-5 dodecahedral honeycomb is one of four compact regular space-filling tessellations (or honeycombs) in hyperbolic 3-space. With Schläfli symbol {5,3,5}, it has five dodecahedral cells around each edge, and each vertex is surrounded by twenty dodecahedra. Its vertex figure is a regular icosahedron. {{Honeycomb}}DescriptionThe 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 60°, for an ideal hyperbolic regular dodecahedron with infinitely long edges. The dodecahedra in this dodecahedral honeycomb are sized so that all of their dihedral angles are exactly 72°. ImagesRelated polytopes and honeycombsThere are four regular compact honeycombs in 3D hyperbolic space: {{Regular compact H3 honeycombs}}There is another honeycomb in hyperbolic 3-space called the order-4 dodecahedral honeycomb, {5,3,4}, which has only four dodecahedra per edge. These honeycombs are also related to the 120-cell which can be considered as a honeycomb in positively curved space (the surface of a 4-dimensional sphere), with three dodecahedra on each edge, {5,3,3}. Lastly the dodecahedral ditope, {5,3,2} exists on a 3-sphere, with 2 hemispherical cells. There are nine uniform honeycombs in the [5,3,5] Coxeter group family, including this regular form. Also the bitruncated form, t1,2{5,3,5}, {{CDD|node|5|node_1|3|node_1|5|node}}, of this honeycomb has all truncated icosahedron cells. {{535 family}}The Seifert–Weber space is a compact manifold that can be formed as a quotient space of the order-5 dodecahedral honeycomb. This honeycomb is a part of a sequence of polychora and honeycombs with icosahedron vertex figures: {{Icosahedral vertex figure tessellations}}This honeycomb is a part of a sequence of regular polytopes and honeycombs with dodecahedral cells: {{Dodecahedral_tessellations small}}{{Symmetric2_tessellations}}Rectified order-5 dodecahedral honeycomb
The rectified order-5 dodecahedral honeycomb, {{CDD|node|5|node_1|3|node|5|node}}, has alternating icosahedron and icosidodecahedron cells, with a pentagonal prism vertex figure. Related tilings and honeycombThere are four rectified compact regular honeycombs: {{Rectified compact H3 honeycombs}}{{Pentagonal prism vertex figure tessellations}}{{-}}Truncated order-5 dodecahedral honeycomb
The truncated order-5 dodecahedral honeycomb, {{CDD|node_1|5|node_1|3|node|5|node}}, has icosahedron and truncated dodecahedron cells, with a pentagonal pyramid vertex figure. Related honeycombs{{Truncated compact H3 honeycombs}}{{-}}Bitruncated order-5 dodecahedral honeycomb
The bitruncated order-5 dodecahedral honeycomb, {{CDD|node|5|node_1|3|node_1|5|node}}, has truncated icosahedron cells, with a disphenoid vertex figure. Related honeycombs{{Bitruncated compact H3 honeycombs}}{{-}}Cantellated order-5 dodecahedral honeycomb
The cantellated order-5 dodecahedral honeycomb, {{CDD|node_1|5|node|3|node_1|5|node}}, has alternating rhombicosidodecahedron and icosidodecahedron cells, with a triangular prism vertex figure. Related honeycombs{{Cantellated compact H3 honeycombs}}{{-}}Cantitruncated order-5 dodecahedral honeycomb
The cantitruncated order-5 dodecahedral honeycomb, {{CDD|node_1|5|node_1|3|node_1|5|node}}, has truncated icosidodecahedron, icosidodecahedron, and pentagonal prism cells, with a mirrored sphenoid vertex figure. Related honeycombs{{Cantitruncated compact H3 honeycombs}}{{-}}Runcinated order-5 dodecahedral honeycomb
The runcinated order-5 dodecahedral honeycomb, {{CDD|node_1|5|node|3|node|5|node_1}}, has dodecahedron and pentagonal prism cells, with a triangular antiprism vertex figure. Related honeycombs{{Runcinated compact H3 honeycombs}}{{-}}Runcitruncated order-5 dodecahedral honeycomb
The runcitruncated order-5 dodecahedral honeycomb, {{CDD|node_1|5|node_1|3|node|5|node_1}}, has truncated dodecahedron, icosidodecahedron and pentagonal prism cells, with a distorted square pyramid vertex figure. Related honeycombs{{Runcitruncated compact H3 honeycombs}}{{-}}Omnitruncated order-5 dodecahedral honeycomb
The omnitruncated order-5 dodecahedral honeycomb, {{CDD|node_1|5|node_1|3|node_1|5|node_1}}, has truncated icosidodecahedron and decagonal prism cells, with a disphenoid vertex figure. Related honeycombs{{Omnitruncated compact H3 honeycombs}}{{-}}See also
References
2 : Honeycombs (geometry)|Self-dual tilings |
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