Английская Википедия:Hexagonal tiling honeycomb
| Hexagonal tiling honeycomb | |
|---|---|
| Файл:H3 633 FC boundary.png Perspective projection view within Poincaré disk model | |
| Type | Hyperbolic regular honeycomb Paracompact uniform honeycomb |
| Schläfli symbols | {6,3,3} t{3,6,3} 2t{6,3,6} 2t{6,3[3]} t{3[3,3]} |
| Coxeter diagrams | Шаблон:CDD Шаблон:CDD Шаблон:CDD Шаблон:CDD Шаблон:CDD ↔ Шаблон:CDD ↔ Шаблон:CDD ↔ Шаблон:CDD ↔ Шаблон:CDD |
| Cells | {6,3} Файл:Uniform tiling 63-t0.png |
| Faces | hexagon {6} |
| Edge figure | triangle {3} |
| Vertex figure | Файл:Order-3 hexagonal tiling honeycomb verf.png tetrahedron {3,3} |
| Dual | Order-6 tetrahedral honeycomb |
| Coxeter groups | <math>{\overline{V}}_3</math>, [3,3,6] <math>{\overline{Y}}_3</math>, [3,6,3] <math>{\overline{Z}}_3</math>, [6,3,6] <math>{\overline{VP}}_3</math>, [6,3[3]] <math>{\overline{PP}}_3</math>, [3[3,3]] |
| Properties | Regular |
In the field of hyperbolic geometry, the hexagonal tiling honeycomb is one of 11 regular paracompact honeycombs in 3-dimensional hyperbolic space. It is paracompact because it has cells composed of an infinite number of faces. Each cell is a hexagonal tiling whose vertices lie on a horosphere, a surface in hyperbolic space that approaches a single ideal point at infinity.
The Schläfli symbol of the hexagonal tiling honeycomb is {6,3,3}. Since that of the hexagonal tiling is {6,3}, this honeycomb has three such hexagonal tilings meeting at each edge. Since the Schläfli symbol of the tetrahedron is {3,3}, the vertex figure of this honeycomb is a tetrahedron. Thus, four hexagonal tilings meet at each vertex of this honeycomb, six hexagons meet at each vertex, and four edges meet at each vertex.[1]
Images
Viewed in perspective outside of a Poincaré disk model, the image above shows one hexagonal tiling cell within the honeycomb, and its mid-radius horosphere (the horosphere incident with edge midpoints). In this projection, the hexagons grow infinitely small towards the infinite boundary, asymptoting towards a single ideal point. It can be seen as similar to the order-3 apeirogonal tiling, {∞,3} of H2, with horocycles circumscribing vertices of apeirogonal faces.
| {6,3,3} | {∞,3} |
|---|---|
| Файл:633 honeycomb one cell horosphere.png | Файл:Order-3 apeirogonal tiling one cell horocycle.png |
| One hexagonal tiling cell of the hexagonal tiling honeycomb | An order-3 apeirogonal tiling with a green apeirogon and its horocycle |
Symmetry constructions
It has a total of five reflectional constructions from five related 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.
Related polytopes and honeycombs
The hexagonal tiling honeycomb is a regular hyperbolic honeycomb in 3-space, and one of 11 which are paracompact. Шаблон:Regular paracompact H3 honeycombs
It is one of 15 uniform paracompact honeycombs in the [6,3,3] Coxeter group, along with its dual, the order-6 tetrahedral honeycomb. Шаблон:633 family
It is part of a sequence of regular polychora, which include the 5-cell {3,3,3}, tesseract {4,3,3}, and 120-cell {5,3,3} of Euclidean 4-space, along with other hyperbolic honeycombs containing tetrahedral vertex figures. Шаблон:Tetrahedral vertex figure tessellations It is also part of a sequence of regular honeycombs of the form {6,3,p}, which are each composed of hexagonal tiling cells: Шаблон:Hexagonal tiling cell tessellations
Rectified hexagonal tiling honeycomb
| Rectified hexagonal tiling honeycomb | |
|---|---|
| Type | Paracompact uniform honeycomb |
| Schläfli symbols | r{6,3,3} or t1{6,3,3} |
| Coxeter diagrams | Шаблон:CDD Шаблон:CDD ↔ Шаблон:CDD |
| Cells | {3,3} Файл:Uniform polyhedron-33-t2.png r{6,3} Файл:Uniform tiling 63-t1.png or Файл:Uniform tiling 333-t12.png |
| Faces | triangle {3} hexagon {6} |
| Vertex figure | Файл:Rectified order-3 hexagonal tiling honeycomb verf.png triangular prism |
| Coxeter groups | <math>{\overline{V}}_3</math>, [3,3,6] <math>{\overline{P}}_3</math>, [3,3[3]] |
| Properties | Vertex-transitive, edge-transitive |
The rectified hexagonal tiling honeycomb, t1{6,3,3}, Шаблон:CDD has tetrahedral and trihexagonal tiling facets, with a triangular prism vertex figure. The Шаблон:CDD half-symmetry construction alternates two types of tetrahedra.
| Hexagonal tiling honeycomb Шаблон:CDD |
Rectified hexagonal tiling honeycomb Шаблон:CDD or Шаблон:CDD |
|---|---|
| Файл:Hyperbolic 3d hexagonal tiling.png | Файл:Hyperbolic 3d rectified hexagonal tiling.png |
| Related H2 tilings | |
| Order-3 apeirogonal tiling Шаблон:CDD |
Triapeirogonal tiling Шаблон:CDD or Шаблон:CDD |
| Файл:H2-I-3-dual.svg | Файл:H2 tiling 23i-2.pngФайл:H2 tiling 33i-3.png |
Truncated hexagonal tiling honeycomb
| Truncated hexagonal tiling honeycomb | |
|---|---|
| Type | Paracompact uniform honeycomb |
| Schläfli symbol | t{6,3,3} or t0,1{6,3,3} |
| Coxeter diagram | Шаблон:CDD |
| Cells | {3,3} Файл:Uniform polyhedron-33-t2.png t{6,3} Файл:Uniform tiling 63-t01.png |
| Faces | triangle {3} dodecagon {12} |
| Vertex figure | Файл:Truncated order-3 hexagonal tiling honeycomb verf.png triangular pyramid |
| Coxeter groups | <math>{\overline{V}}_3</math>, [3,3,6] |
| Properties | Vertex-transitive |
The truncated hexagonal tiling honeycomb, t0,1{6,3,3}, Шаблон:CDD has tetrahedral and truncated hexagonal tiling facets, with a triangular pyramid vertex figure.
It is similar to the 2D hyperbolic truncated order-3 apeirogonal tiling, t{∞,3} with apeirogonal and triangle faces:
Bitruncated hexagonal tiling honeycomb
| Bitruncated hexagonal tiling honeycomb Bitruncated order-6 tetrahedral honeycomb | |
|---|---|
| Type | Paracompact uniform honeycomb |
| Schläfli symbol | 2t{6,3,3} or t1,2{6,3,3} |
| Coxeter diagram | Шаблон:CDD Шаблон:CDD ↔ Шаблон:CDD |
| Cells | t{3,3} Файл:Uniform polyhedron-33-t01.png t{3,6} Файл:Uniform tiling 63-t12.png |
| Faces | triangle {3} hexagon {6} |
| Vertex figure | Файл:Bitruncated order-3 hexagonal tiling honeycomb verf.png digonal disphenoid |
| Coxeter groups | <math>{\overline{V}}_3</math>, [3,3,6] <math>{\overline{P}}_3</math>, [3,3[3]] |
| Properties | Vertex-transitive |
The bitruncated hexagonal tiling honeycomb or bitruncated order-6 tetrahedral honeycomb, t1,2{6,3,3}, Шаблон:CDD has truncated tetrahedron and hexagonal tiling cells, with a digonal disphenoid vertex figure.
Cantellated hexagonal tiling honeycomb
| Cantellated hexagonal tiling honeycomb | |
|---|---|
| Type | Paracompact uniform honeycomb |
| Schläfli symbol | rr{6,3,3} or t0,2{6,3,3} |
| Coxeter diagram | Шаблон:CDD |
| Cells | r{3,3} Файл:Uniform polyhedron-33-t1.png rr{6,3} Файл:Uniform tiling 63-t02.png {}×{3} Файл:Triangular prism.png |
| Faces | triangle {3} square {4} hexagon {6} |
| Vertex figure | Файл:Cantellated order-3 hexagonal tiling honeycomb verf.png wedge |
| Coxeter groups | <math>{\overline{V}}_3</math>, [3,3,6] |
| Properties | Vertex-transitive |
The cantellated hexagonal tiling honeycomb, t0,2{6,3,3}, Шаблон:CDD has octahedron, rhombitrihexagonal tiling, and triangular prism cells, with a wedge vertex figure.
Cantitruncated hexagonal tiling honeycomb
| Cantitruncated hexagonal tiling honeycomb | |
|---|---|
| Type | Paracompact uniform honeycomb |
| Schläfli symbol | tr{6,3,3} or t0,1,2{6,3,3} |
| Coxeter diagram | Шаблон:CDD |
| Cells | t{3,3} Файл:Uniform polyhedron-33-t01.png tr{6,3} Файл:Uniform tiling 63-t012.svg {}×{3} Файл:Triangular prism.png |
| Faces | triangle {3} square {4} hexagon {6} dodecagon {12} |
| Vertex figure | Файл:Cantitruncated order-3 hexagonal tiling honeycomb verf.png mirrored sphenoid |
| Coxeter groups | <math>{\overline{V}}_3</math>, [3,3,6] |
| Properties | Vertex-transitive |
The cantitruncated hexagonal tiling honeycomb, t0,1,2{6,3,3}, Шаблон:CDD has truncated tetrahedron, truncated trihexagonal tiling, and triangular prism cells, with a mirrored sphenoid vertex figure.
Runcinated hexagonal tiling honeycomb
| Runcinated hexagonal tiling honeycomb | |
|---|---|
| Type | Paracompact uniform honeycomb |
| Schläfli symbol | t0,3{6,3,3} |
| Coxeter diagram | Шаблон:CDD |
| Cells | {3,3} Файл:Uniform polyhedron-33-t0.png {6,3} Файл:Uniform tiling 63-t0.png {}×{6}Файл:Hexagonal prism.png {}×{3} Файл:Triangular prism.png |
| Faces | triangle {3} square {4} hexagon {6} |
| Vertex figure | Файл:Runcinated order-3 hexagonal tiling honeycomb verf.png irregular triangular antiprism |
| Coxeter groups | <math>{\overline{V}}_3</math>, [3,3,6] |
| Properties | Vertex-transitive |
The runcinated hexagonal tiling honeycomb, t0,3{6,3,3}, Шаблон:CDD has tetrahedron, hexagonal tiling, hexagonal prism, and triangular prism cells, with an irregular triangular antiprism vertex figure.
Runcitruncated hexagonal tiling honeycomb
| Runcitruncated hexagonal tiling honeycomb | |
|---|---|
| Type | Paracompact uniform honeycomb |
| Schläfli symbol | t0,1,3{6,3,3} |
| Coxeter diagram | Шаблон:CDD |
| Cells | rr{3,3} Файл:Uniform polyhedron-33-t02.png {}x{3} Файл:Triangular prism.png {}x{12} Файл:Dodecagonal prism.png t{6,3} Файл:Uniform tiling 63-t01.png |
| Faces | triangle {3} square {4} dodecagon {12} |
| Vertex figure | Файл:Runcitruncated order-3 hexagonal tiling honeycomb verf.png isosceles-trapezoidal pyramid |
| Coxeter groups | <math>{\overline{V}}_3</math>, [3,3,6] |
| Properties | Vertex-transitive |
The runcitruncated hexagonal tiling honeycomb, t0,1,3{6,3,3}, Шаблон:CDD has cuboctahedron, triangular prism, dodecagonal prism, and truncated hexagonal tiling cells, with an isosceles-trapezoidal pyramid vertex figure.
Runcicantellated hexagonal tiling honeycomb
| Runcicantellated hexagonal tiling honeycomb runcitruncated order-6 tetrahedral honeycomb | |
|---|---|
| Type | Paracompact uniform honeycomb |
| Schläfli symbol | t0,2,3{6,3,3} |
| Coxeter diagram | Шаблон:CDD |
| Cells | t{3,3} Файл:Uniform polyhedron-33-t12.png {}x{6} Файл:Hexagonal prism.png rr{6,3} Файл:Uniform tiling 63-t02.png |
| Faces | triangle {3} square {4} hexagon {6} |
| Vertex figure | Файл:Runcitruncated order-6 tetrahedral honeycomb verf.png isosceles-trapezoidal pyramid |
| Coxeter groups | <math>{\overline{V}}_3</math>, [3,3,6] |
| Properties | Vertex-transitive |
The runcicantellated hexagonal tiling honeycomb or runcitruncated order-6 tetrahedral honeycomb, t0,2,3{6,3,3}, Шаблон:CDD has truncated tetrahedron, hexagonal prism, and rhombitrihexagonal tiling cells, with an isosceles-trapezoidal pyramid vertex figure.
Omnitruncated hexagonal tiling honeycomb
| Omnitruncated hexagonal tiling honeycomb Omnitruncated order-6 tetrahedral honeycomb | |
|---|---|
| Type | Paracompact uniform honeycomb |
| Schläfli symbol | t0,1,2,3{6,3,3} |
| Coxeter diagram | Шаблон:CDD |
| Cells | tr{3,3} Файл:Uniform polyhedron-33-t012.png {}x{6} Файл:Hexagonal prism.png {}x{12} Файл:Dodecagonal prism.png tr{6,3} Файл:Uniform tiling 63-t012.svg |
| Faces | square {4} hexagon {6} dodecagon {12} |
| Vertex figure | Файл:Omnitruncated order-3 hexagonal tiling honeycomb verf.png irregular tetrahedron |
| Coxeter groups | <math>{\overline{V}}_3</math>, [3,3,6] |
| Properties | Vertex-transitive |
The omnitruncated hexagonal tiling honeycomb or omnitruncated order-6 tetrahedral honeycomb, t0,1,2,3{6,3,3}, Шаблон:CDD has truncated octahedron, hexagonal prism, dodecagonal prism, and truncated trihexagonal tiling cells, with an irregular tetrahedron vertex figure.
See also
- Convex uniform honeycombs in hyperbolic space
- Regular tessellations of hyperbolic 3-space
- Paracompact uniform honeycombs
- Alternated 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]
External links
- John Baez, Visual Insight: {6,3,3} Honeycomb (2014/03/15)
- John Baez, Visual Insight: {6,3,3} Honeycomb in Upper Half Space (2013/09/15)
- John Baez, Visual Insight: Truncated {6,3,3} Honeycomb (2016/12/01)
- ↑ Coxeter The Beauty of Geometry, 1999, Chapter 10, Table III
