Английская Википедия:Alternated hexagonal tiling honeycomb
| Alternated hexagonal tiling honeycomb | |
|---|---|
| Type | Paracompact uniform honeycomb Semiregular honeycomb |
| Schläfli symbols | h{6,3,3} s{3,6,3} 2s{6,3,6} 2s{6,3[3]} s{3[3,3]} |
| Coxeter diagrams | Шаблон:CDD ↔ Шаблон:CDD Шаблон:CDD Шаблон:CDD Шаблон:CDD ↔ Шаблон:CDD Шаблон:CDD ↔ Шаблон:CDD ↔ Шаблон:CDD |
| Cells | {3,3} Файл:Uniform polyhedron-33-t0.png {3[3]} Файл:Uniform tiling 333-t0.png |
| Faces | triangle {3} |
| Vertex figure | Файл:Uniform polyhedron-33-t01.png Шаблон:CDD truncated tetrahedron |
| Coxeter groups | <math>{\overline{P}}_3</math>, [3,3[3]] 1/2 <math>{\overline{V}}_3</math>, [6,3,3] 1/2 <math>{\overline{Y}}_3</math>, [3,6,3] 1/2 <math>{\overline{Z}}_3</math>, [6,3,6] 1/2 <math>{\overline{VP}}_3</math>, [6,3[3]] 1/2 <math>{\overline{PP}}_3</math>, [3[3,3]] |
| Properties | Vertex-transitive, edge-transitive, quasiregular |
In three-dimensional hyperbolic geometry, the alternated hexagonal tiling honeycomb, h{6,3,3}, Шаблон:CDD or Шаблон:CDD, is a semiregular tessellation with tetrahedron and triangular tiling cells arranged in an octahedron vertex figure. It is named after its construction, as an alteration of a hexagonal tiling honeycomb.
Symmetry constructions
It has five alternated constructions from reflectional Coxeter groups all with four mirrors and only the first being regular: Шаблон:CDD [6,3,3], Шаблон:CDD [3,6,3], Шаблон:CDD [6,3,6], Шаблон:CDD [6,3[3]] and [3[3,3]] Шаблон:CDD, having 1, 4, 6, 12 and 24 times larger fundamental domains respectively. In Coxeter notation subgroup markups, they are related as: [6,(3,3)*] (remove 3 mirrors, index 24 subgroup); [3,6,3*] or [3*,6,3] (remove 2 mirrors, index 6 subgroup); [1+,6,3,6,1+] (remove two orthogonal mirrors, index 4 subgroup); all of these are isomorphic to [3[3,3]]. The ringed Coxeter diagrams are Шаблон:CDD, Шаблон:CDD, Шаблон:CDD, Шаблон:CDD and Шаблон:CDD, representing different types (colors) of hexagonal tilings in the Wythoff construction. Шаблон:-
Related honeycombs
The alternated hexagonal tiling honeycomb has 3 related forms: the cantic hexagonal tiling honeycomb, Шаблон:CDD; the runcic hexagonal tiling honeycomb, Шаблон:CDD; and the runcicantic hexagonal tiling honeycomb, Шаблон:CDD.
Cantic hexagonal tiling honeycomb
| Cantic hexagonal tiling honeycomb | |
|---|---|
| Type | Paracompact uniform honeycomb |
| Schläfli symbols | h2{6,3,3} |
| Coxeter diagrams | Шаблон:CDD ↔ Шаблон:CDD |
| Cells | r{3,3} Файл:Uniform polyhedron-33-t1.png t{3,3} Файл:Uniform polyhedron-33-t01.png h2{6,3} Файл:Uniform tiling 333-t01.png |
| Faces | triangle {3} hexagon {6} |
| Vertex figure | Файл:Cantic hexagonal tiling honeycomb verf.png wedge |
| Coxeter groups | <math>{\overline{P}}_3</math>, [3,3[3]] |
| Properties | Vertex-transitive |
The cantic hexagonal tiling honeycomb, h2{6,3,3}, Шаблон:CDD or Шаблон:CDD, is composed of octahedron, truncated tetrahedron, and trihexagonal tiling facets, with a wedge vertex figure.
Runcic hexagonal tiling honeycomb
| Runcic hexagonal tiling honeycomb | |
|---|---|
| Type | Paracompact uniform honeycomb |
| Schläfli symbols | h3{6,3,3} |
| Coxeter diagrams | Шаблон:CDD ↔ Шаблон:CDD |
| Cells | {3,3} Файл:Uniform polyhedron-33-t0.png {}x{3} Файл:Triangular prism.png rr{3,3} Файл:Uniform polyhedron-33-t02.png {3[3]} Файл:Uniform tiling 333-t0.png |
| Faces | triangle {3} square {4} hexagon {6} |
| Vertex figure | Файл:Runcic hexagonal tiling honeycomb verf.png triangular cupola |
| Coxeter groups | <math>{\overline{P}}_3</math>, [3,3[3]] |
| Properties | Vertex-transitive |
The runcic hexagonal tiling honeycomb, h3{6,3,3}, Шаблон:CDD or Шаблон:CDD, has tetrahedron, triangular prism, cuboctahedron, and triangular tiling facets, with a triangular cupola vertex figure. Шаблон:-
Runcicantic hexagonal tiling honeycomb
| Runcicantic hexagonal tiling honeycomb | |
|---|---|
| Type | Paracompact uniform honeycomb |
| Schläfli symbols | h2,3{6,3,3} |
| Coxeter diagrams | Шаблон:CDD ↔ Шаблон:CDD |
| Cells | t{3,3} Файл:Uniform polyhedron-33-t01.png {}x{3} Файл:Triangular prism.png tr{3,3} Файл:Uniform polyhedron-33-t012.png h2{6,3} Файл:Uniform tiling 333-t01.png |
| Faces | triangle {3} square {4} hexagon {6} |
| Vertex figure | Файл:Runcicantic hexagonal tiling honeycomb verf.png rectangular pyramid |
| Coxeter groups | <math>{\overline{P}}_3</math>, [3,3[3]] |
| Properties | Vertex-transitive |
The runcicantic hexagonal tiling honeycomb, h2,3{6,3,3}, Шаблон:CDD or Шаблон:CDD, has truncated tetrahedron, triangular prism, truncated octahedron, and trihexagonal tiling facets, with a rectangular pyramid vertex figure.
See also
- Convex uniform honeycombs in hyperbolic space
- Regular tessellations of hyperbolic 3-space
- Paracompact uniform honeycombs
- Semiregular honeycomb
- Hexagonal tiling honeycomb
References
- Coxeter, Regular Polytopes, 3rd. ed., Dover Publications, 1973. Шаблон:ISBN. (Tables I and II: Regular polytopes and honeycombs, pp. 294–296)
- The Beauty of Geometry: Twelve Essays (1999), Dover Publications, Шаблон:LCCN, Шаблон:ISBN (Chapter 10, Regular Honeycombs in Hyperbolic Space Шаблон:Webarchive) Table III
- Jeffrey R. Weeks The Shape of Space, 2nd edition Шаблон:ISBN (Chapters 16–17: Geometries on Three-manifolds I,II)
- N. W. Johnson, R. Kellerhals, J. G. Ratcliffe, S. T. Tschantz, The size of a hyperbolic Coxeter simplex, Transformation Groups (1999), Volume 4, Issue 4, pp 329–353 [1] [2]
- N. W. Johnson, R. Kellerhals, J. G. Ratcliffe, S. T. Tschantz, Commensurability classes of hyperbolic Coxeter groups, (2002) H3: p130. [3]
