Order-5 hexagonal tiling honeycomb
| Order-5 hexagonal tiling honeycomb | |
|---|---|
 Perspective projection view from center of Poincaré disk model |
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| Type | Hyperbolic regular honeycomb Paracompact uniform honeycomb |
| Schläfli symbol | {6,3,5} |
| Coxeter-Dynkin diagram |      |
| Cells | {6,3}  |
| Faces | hexagon {6} |
| Edge figure | pentagon {5} |
| Vertex figure | {3,5} |
| Dual | Order-6 dodecahedral honeycomb |
| Coxeter group | HV3, [6,3,5] |
| Properties | Regular |
In the field of hyperbolic geometry, the order-5 hexagonal tiling honeycomb arises one of 11 regular paracompact honeycombs in 3-dimensional hyperbolic space. It is called paracompact because it has infinite cells. Each cell consists of a hexagonal tiling whose vertices lie on a horosphere: a flat plane in hyperbolic space that approaches a single ideal point at infinity.
The Schläfli symbol of the order-5 hexagonal tiling honeycomb is {6,3,5}. Since that of the hexagonal tiling of the plane is {6,3}, this honeycomb has five such hexagonal tilings meeting at each edge. Since the Schläfli symbol of the icosahedron is {3,5}, the vertex figure of this honeycomb is an icosahedron. Thus, 20 hexagonal tilings meet at each vertex of this honeycomb.[1]
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.
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Symmetry[edit]
A lower symmetry, [6,(3,5)*], index 120 construction exists with regular dodecahedral fundamental domains, and a icosahedral shaped Coxeter diagram with 6 axial infinite order (ultraparallel) branches.
Images[edit]
It is similar to the 2D hyperbolic regular tiling, {∞,5}, with infinite apeirogonal faces, and 5 meeting around every vertex (peak).
Related polytopes and honeycombs[edit]
It is one of 15 regular hyperbolic honeycombs in 3-space, 11 of which like this one are paracompact, with infinite cells or vertex figures.
 {6,3,3} |
 {6,3,4} |
{6,3,5} |
 {6,3,6} |
 {4,4,3} |
 {4,4,4} |
 {3,3,6} |
 {4,3,6} |
 {5,3,6} |
 {3,6,3} |
 {3,4,4} |
There are 15 uniform honeycombs in the [5,3,6] Coxeter group family, including this regular form and its regular dual, order-5 hexagonal tiling honeycomb, {6,3,5}.
| {6,3,5} | t1{6,3,5} | t0,1{6,3,5} | t0,2{6,3,5} | t0,3{6,3,5} | t0,1,2{6,3,5} | t0,1,3{6,3,5} | t0,1,2,3{6,3,5} |
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| {5,3,6} | t1{5,3,6} | t0,1{5,3,6} | t0,2{5,3,6} | t1,2{5,3,6} | t0,1,2{5,3,6} | t0,1,3{5,3,6} | t0,1,2,3{5,3,6} |
It has a related alternation honeycomb, represented by 
↔ 
, having icosahedron and triangular tiling cells.
It is a part of sequence of regular honeycombs with hexagonal tiling hyperbolic honeycombs of the form {6,3,p}:
| Space | H3 | ||||||
|---|---|---|---|---|---|---|---|
| Form | Paracompact | Noncompact | |||||
| Name | {6,3,3} | {6,3,4} | {6,3,5} | {6,3,6} | {6,3,7} |
{6,3,8} |
... {6,3,∞} |
Coxeter    |
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| Vertex figure {3,p}  |
 {3,3} |
 {3,4} |
 {3,5} |
 {3,6} |
 {3,7} |
 {3,8} |
 {3,∞} |
This honeycomb is a part of a sequence of polychora and honeycombs with icosahedron vertex figures:
| Space | S3 | H3 | ||||
|---|---|---|---|---|---|---|
| Form | Finite | Compact | Paracompact | Noncompact | ||
| Name | {3,3,5} |
{4,3,5} |
{5,3,5} |
{6,3,5} |
{7,3,5} |
... {∞,3,5} |
| Image |  |
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| Cells | {3,3} |
 {4,3} |
 {5,3} |
{6,3} |
 {∞,3} |
 {∞,3} |
Rectified order-5 hexagonal tiling honeycomb[edit]
| Rectified order-5 hexagonal tiling honeycomb | |
|---|---|
| Type | Paracompact uniform honeycomb |
| Schläfli symbols | r{6,3,5} or t1{6,3,5} 2r{5,3[3]} |
| Coxeter diagrams | ↔ |
| Cells | {3,5}  r{6,3}, r{3[3]}   |
| Faces | Triangle {3} Pentagon {5} Hexagon {6} |
| Vertex figure |  Pentagonal prism {}×{5} |
| Coxeter groups |  , [6,3,5] |
| Properties | Vertex-transitive, edge-transitive |
The rectified order-5 hexagonal tiling honeycomb, t1{6,3,5}, has icosahedron and trihexagonal tiling facets, with a pentagonal prism vertex figure.
It is similar to the 2D hyperbolic infinite-order square tiling, r{∞,5} with pentagon and apeirogonal faces. All vertices are on the ideal surface.
| Space | S3 | H3 | ||||
|---|---|---|---|---|---|---|
| Form | Finite | Compact | Paracompact | Noncompact | ||
| Name | r{3,3,5} |
r{4,3,5} |
r{5,3,5} |
r{6,3,5} |
r{7,3,5} |
... r{∞,3,5} |
| Image |  |
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| Cells {3,5} |
 r{3,3} |
 r{4,3} |
 r{5,3} |
r{6,3} |
 r{7,3} |
 r{∞,3} |
Truncated order-5 hexagonal tiling honeycomb[edit]
| Truncated order-5 hexagonal tiling honeycomb | |
|---|---|
| Type | Paracompact uniform honeycomb |
| Schläfli symbol | t{6,3,5} or t0,1{6,3,5} |
| Coxeter diagram | |
| Cells | {3,5} t{6,3}  |
| Faces | Triangle {3} Pentagon {5} Hexagon {6} |
| Vertex figure |  Pentagonal pyramid {}v{5} |
| Coxeter groups | , [6,3,5] |
| Properties | Vertex-transitive |
The truncated order-5 hexagonal tiling honeycomb, t0,1{6,3,5}, has icosahedron and triangular tiling facets, with a pentagonal pyramid vertex figure.
Cantellated order-5 hexagonal tiling honeycomb[edit]
| Cantellated order-5 hexagonal tiling honeycomb | |
|---|---|
| Type | Paracompact uniform honeycomb |
| Schläfli symbol | rr{6,3,5} or t0,2{6,3,5} |
| Coxeter diagram | |
| Cells | r{3,5}  rr{6,3}  |
| Faces | Triangle {3} Pentagon {5} Hexagon {6} |
| Vertex figure |  triangular prism |
| Coxeter groups | , [6,3,5] |
| Properties | Vertex-transitive |
The cantellated order-5 hexagonal tiling honeycomb, t0,2{6,3,5}, has icosidodecahedron and rhombitrihexagonal tiling facets, with a triangular prism vertex figure.
See also[edit]
References[edit]
- ^ Coxeter The Beauty of Geometry, 1999, Chapter 10, Table III
- Coxeter, Regular Polytopes, 3rd. ed., Dover Publications, 1973. ISBN 0-486-61480-8. (Tables I and II: Regular polytopes and honeycombs, pp. 294–296)
- The Beauty of Geometry: Twelve Essays (1999), Dover Publications, LCCN 99-35678, ISBN 0-486-40919-8 (Chapter 10, Regular Honeycombs in Hyperbolic Space) Table III
- 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, (2015) Chapter 13: Hyperbolic Coxeter groups
