| 释义 |
- Description
- Symmetry
- Images
- Related polytopes and honeycombs Rectified order-4 dodecahedral honeycomb Related honeycombs Truncated order-4 dodecahedral honeycomb Related honeycombs Bitruncated order-4 dodecahedral honeycomb Related honeycombs Cantellated order-4 dodecahedral honeycomb Related honeycombs Cantitruncated order-4 dodecahedral honeycomb Related honeycombs Runcitruncated order-4 dodecahedral honeycomb Related honeycombs
- See also
- References
| Order-4 dodecahedral honeycomb | | | Type | Hyperbolic regular honeycomb | | Schläfli symbol | {5,3,4}
{5,31,1} | | Coxeter diagram | node_1|5|node|3|node|4|node}}
{{CDD|node_1|5|node|3|node|4|node_h0}} ↔ {{CDD|node_1|5|node|split1|nodes}} | | Cells | {5,3} | | Faces | pentagon {5} | | Edge figure | square {4} | | Vertex figure |
octahedron | | Dual | Order-5 cubic honeycomb | | Coxeter group | BH}}3, [5,3,4]
{{overline|DH}}3, [5,31,1] | | Properties | Regular, Quasiregular honeycomb |
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. {{Honeycomb}}DescriptionThe 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 is a half symmetry construction, {5,31,1}, with two types (colors) of dodecahedra in the Wythoff construction. {{CDD|node_1|5|node|3|node|4|node_h0}} ↔ {{CDD|node_1|5|node|split1|nodes}}. Images
Beltrami-Klein model Related polytopes and honeycombs There are four regular compact honeycombs in 3D hyperbolic space: {{Regular compact H3 honeycombs}}There are fifteen uniform honeycombs in the [5,3,4] Coxeter group family, including this regular form. {{534 family}}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: {{Octahedral_vertex_figure_tessellations}}This honeycomb is a part of a sequence of polychora and honeycombs with dodecahedral cells: {{Dodecahedral_cell_tessellations}} Rectified order-4 dodecahedral honeycomb | Rectified order-4 dodecahedral honeycomb | | Type | Uniform honeycombs in hyperbolic space | | Schläfli symbol | r{5,3,4}
r{5,31,1} | | Coxeter diagram | node|5|node_1|3|node|4|node}}
{{CDD|node|5|node_1|3|node|4|node_h0}} ↔ {{CDD|node|5|node_1|split1|nodes}} | | Cells | r{5,3}
{3,4} | | Faces | triangle {3}
pentagon {5} | | Vertex figure |
cube | | Coxeter group | BH}}3, [5,3,4]
{{overline|DH}}3, [5,31,1] | | Properties | Vertex-transitive, edge-transitive |
The rectified order-4 dodecahedral honeycomb, {{CDD|node|5|node_1|3|node|4|node}}, has alternating octahedron and icosidodecahedron cells, with a cube vertex figure. Related honeycombsThere are four rectified compact regular honeycombs: {{Rectified compact H3 honeycombs}}{{-}} Truncated order-4 dodecahedral honeycomb | Truncated order-4 dodecahedral honeycomb | | Type | Uniform honeycombs in hyperbolic space | | Schläfli symbol | t{5,3,4}
t{5,31,1} | | Coxeter diagram | node_1|5|node_1|3|node|4|node}}
{{CDD|node_1|5|node_1|3|node|4|node_h0}} ↔ {{CDD|node_1|5|node_1|split1|nodes}} | | Cells | t{5,3}
{3,4} | | Faces | triangle {3}
decagon {10} | | Vertex figure |
Square pyramid | | Coxeter group | BH}}3, [5,3,4]
{{overline|DH}}3, [5,31,1] | | Properties | Vertex-transitive |
The truncated order-4 dodecahedral honeycomb, {{CDD|node_1|5|node_1|3|node|4|node}}, 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: Related honeycombs {{Truncated compact H3 honeycombs}}{{-}} Bitruncated order-4 dodecahedral honeycomb Bitruncated order-4 dodecahedral honeycomb
Bitruncated order-5 cubic honeycomb | | Type | Uniform honeycombs in hyperbolic space | | Schläfli symbol | 2t{5,3,4}
2t{5,31,1} | | Coxeter diagram | node|5|node_1|3|node_1|4|node}}
{{CDD|node|5|node_1|3|node_1|4|node_h0}} ↔ {{CDD|node|5|node_1|split1|nodes_11}} | | Cells | t{3,5}
t{3,4} | | Faces | triangle {3}
square {4}
hexagon {6} | | Vertex figure |
tetrahedron | | Coxeter group | BH}}3, [5,3,4]
{{overline|DH}}3, [5,31,1] | | Properties | Vertex-transitive |
The bitruncated order-4 dodecahedral honeycomb, or bitruncated order-5 cubic honeycomb, {{CDD|node_1|5|node_1|3|node|4|node}}, has truncated octahedron and truncated icosahedron cells, with a tetrahedron vertex figure. Related honeycombs{{Bitruncated compact H3 honeycombs}}{{-}} Cantellated order-4 dodecahedral honeycomb | Cantellated order-4 dodecahedral honeycomb | | Type | Uniform honeycombs in hyperbolic space | | Schläfli symbol | rr{5,3,4}
rr{5,31,1} | | Coxeter diagram | node_1|5|node|3|node_1|4|node}}
{{CDD|node_1|5|node|3|node_1|4|node_h0}} ↔ {{CDD|node_1|5|node|split1|nodes_11}} | | Cells | rr{3,5}
r{3,4}
{}x{4} cube | | Faces | triangle {3}
square {4}
pentagon {5} | | Vertex figure |
Triangular prism | | Coxeter group | BH}}3, [5,3,4]
{{overline|DH}}3, [5,31,1] | | Properties | Vertex-transitive |
The cantellated order-4 dodecahedral honeycomb,{{CDD|node_1|5|node|3|node_1|4|node}}, has rhombicosidodecahedron and cuboctahedron, and cube cells, with a triangular prism vertex figure. Related honeycombs{{Cantellated compact H3 honeycombs}}{{-}} Cantitruncated order-4 dodecahedral honeycomb | Cantitruncated order-4 dodecahedral honeycomb | | Type | Uniform honeycombs in hyperbolic space | | Schläfli symbol | tr{5,3,4}
tr{5,31,1} | | Coxeter diagram | node_1|5|node_1|3|node_1|4|node}}
{{CDD|node_1|5|node_1|3|node_1|4|node_h0}} ↔ {{CDD|node_1|5|node_1|split1|nodes_11}} | | Cells | tr{3,5}
t{3,4}
{}x{4} cube | | Faces | square {4}
hexagon {6}
decagon {10} | | Vertex figure |
mirrored sphenoid | | Coxeter group | BH}}3, [5,3,4]
{{overline|DH}}3, [5,31,1] | | Properties | Vertex-transitive |
The cantitruncated order-4 dodecahedral honeycomb, is a uniform honeycomb constructed with a {{CDD|node_1|5|node_1|3|node_1|4|node}} coxeter diagram, and mirrored sphenoid vertex figure. Related honeycombs{{Cantitruncated compact H3 honeycombs}}{{-}} Runcitruncated order-4 dodecahedral honeycomb | Runcitruncated order-4 dodecahedral honeycomb | | Type | Uniform honeycombs in hyperbolic space | | Schläfli symbol | t0,1,3{5,3,4} | | Coxeter diagram | node_1|5|node_1|3|node|4|node_1}} | | Cells | t{5,3}
rr{3,4}
{}x{10}
{}x{4} | | Faces | triangle {3}
square {4}
decagon {10} | | Vertex figure |
quad pyramid | | Coxeter group | BH}}3, [5,3,4] | | Properties | Vertex-transitive |
The runcititruncated order-4 dodecahedral honeycomb, is a uniform honeycomb constructed with a {{CDD|node_1|5|node_1|3|node|4|node_1}} coxeter diagram, and a quadrilateral pyramid vertex figure. Related honeycombs{{Runcitruncated compact H3 honeycombs}} See also - Convex uniform honeycombs in hyperbolic space
- Poincaré homology sphere Poincaré dodecahedral space
- Seifert–Weber space Seifert–Weber dodecahedral space
- List of regular polytopes
References - Coxeter, Regular Polytopes, 3rd. ed., Dover Publications, 1973. {{isbn|0-486-61480-8}}. (Tables I and II: Regular polytopes and honeycombs, pp. 294–296)
- Coxeter, The Beauty of Geometry: Twelve Essays, Dover Publications, 1999 {{isbn|0-486-40919-8}} (Chapter 10: Regular honeycombs in hyperbolic space, Summary tables II,III,IV,V, p212-213)
- Jeffrey R. Weeks The Shape of Space, 2nd edition {{isbn|0-8247-0709-5}} (Chapter 16-17: Geometries on Three-manifolds I,II)
- Norman Johnson Uniform Polytopes, Manuscript
- N.W. Johnson: The Theory of Uniform Polytopes and Honeycombs, Ph.D. Dissertation, University of Toronto, 1966
- N.W. Johnson: Geometries and Transformations, (2018) Chapter 13: Hyperbolic Coxeter groups
{{DEFAULTSORT:Order-4 Dodecahedral Honeycomb}} 1 : Honeycombs (geometry) |