Order-5 hexagonal tiling honeycomb
Order-5 hexagonal tiling honeycomb | |
---|---|
Perspective projection view
from center of Poincaré disk model | |
Type | Paracompact uniform honeycomb
|
Schläfli symbol | {6,3,5} |
Coxeter-Dynkin diagrams |
↔ |
Cells | {6,3} |
Faces | hexagon {6} |
Edge figure |
pentagon {5} |
Vertex figure | icosahedron |
Dual |
Order-6 dodecahedral honeycomb |
Coxeter group | , [5,3,6] |
Properties | Regular |
In the field of
The Schläfli symbol of the order-5 hexagonal tiling honeycomb is {6,3,5}. Since that of the hexagonal tiling 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
Symmetry
A lower-symmetry construction of index 120, [6,(3,5)*], exists with regular
Images
The order-5 hexagonal tiling honeycomb is similar to the 2D hyperbolic regular paracompact order-5 apeirogonal tiling, {∞,5}, with five apeirogonal faces meeting around every vertex.
Related polytopes and honeycombs
The order-5 hexagonal tiling honeycomb is a regular hyperbolic honeycomb in 3-space, and one of 11 which are paracompact.
11 paracompact regular honeycombs | |||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|
{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
{6,3,5} | r{6,3,5}
|
t{6,3,5}
|
rr{6,3,5}
|
t0,3{6,3,5}
|
tr{6,3,5}
|
t0,1,3{6,3,5}
|
t0,1,2,3{6,3,5}
|
---|---|---|---|---|---|---|---|
{5,3,6} | r{5,3,6}
|
t{5,3,6}
|
rr{5,3,6}
|
2t{5,3,6}
|
tr{5,3,6}
|
t0,1,3{5,3,6}
|
t0,1,2,3{5,3,6}
|
The order-5 hexagonal tiling honeycomb has a related alternation honeycomb, represented by ↔ , with icosahedron and triangular tiling cells.
It is a part of sequence of regular hyperbolic honeycombs of the form {6,3,p}, with hexagonal tiling facets:
{6,3,p} honeycombs | |||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|
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 |
|||||||||||
Image | |||||||||||
Vertex figure {3,p} |
{3,3} |
{3,4} |
{3,5} |
{3,6} |
{3,7} |
{3,8} |
{3,∞} |
It is also part of a sequence of
{p,3,5} polytopes | |||||||
---|---|---|---|---|---|---|---|
Space | S3 | H3 | |||||
Form | Finite | Compact | Paracompact | Noncompact | |||
Name | {3,3,5} |
{4,3,5} |
{5,3,5} |
{6,3,5} |
{7,3,5} |
{8,3,5} |
... {∞,3,5} |
Image | |||||||
Cells | {3,3} |
{4,3} |
{5,3} |
{6,3} |
{7,3} |
{8,3} |
{∞,3} |
Rectified order-5 hexagonal tiling honeycomb
Rectified order-5 hexagonal tiling honeycomb | |
---|---|
Type | Paracompact uniform honeycomb
|
Schläfli symbols | r{6,3,5} or t1{6,3,5} |
Coxeter diagrams |
↔ |
Cells | {3,5} r{6,3} or h2{6,3} |
Faces | triangle {3} hexagon {6} |
Vertex figure | pentagonal prism |
Coxeter groups | , [5,3,6] , [5,3[3]] |
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 | ||||||
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
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} dodecagon {12} |
Vertex figure | pentagonal pyramid |
Coxeter groups | , [5,3,6] |
Properties | Vertex-transitive |
The truncated order-5 hexagonal tiling honeycomb, t0,1{6,3,5}, has icosahedron and truncated hexagonal tiling facets, with a pentagonal pyramid vertex figure.
Bitruncated order-5 hexagonal tiling honeycomb
Bitruncated order-5 hexagonal tiling honeycomb | |
---|---|
Type | Paracompact uniform honeycomb
|
Schläfli symbol | 2t{6,3,5} or t1,2{6,3,5} |
Coxeter diagram |
↔ |
Cells | t{3,6} t{3,5} |
Faces | pentagon {5} hexagon {6} |
Vertex figure | digonal disphenoid
|
Coxeter groups | , [5,3,6] , [5,3[3]] |
Properties | Vertex-transitive |
The bitruncated order-5 hexagonal tiling honeycomb, t1,2{6,3,5}, has
Cantellated order-5 hexagonal tiling honeycomb
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} {}x{5} |
Faces | triangle {3} square {4} pentagon {5} hexagon {6} |
Vertex figure | wedge |
Coxeter groups | , [5,3,6] |
Properties | Vertex-transitive |
The cantellated order-5 hexagonal tiling honeycomb, t0,2{6,3,5}, has icosidodecahedron, rhombitrihexagonal tiling, and pentagonal prism facets, with a wedge vertex figure.
Cantitruncated order-5 hexagonal tiling honeycomb
Cantitruncated order-5 hexagonal tiling honeycomb | |
---|---|
Type | Paracompact uniform honeycomb
|
Schläfli symbol | tr{6,3,5} or t0,1,2{6,3,5} |
Coxeter diagram |
|
Cells | t{3,5} tr{6,3} {}x{5} |
Faces | square {4} pentagon {5} hexagon {6} dodecagon {12} |
Vertex figure | mirrored sphenoid
|
Coxeter groups | , [5,3,6] |
Properties | Vertex-transitive |
The cantitruncated order-5 hexagonal tiling honeycomb, t0,1,2{6,3,5}, has
Runcinated order-5 hexagonal tiling honeycomb
Runcinated order-5 hexagonal tiling honeycomb | |
---|---|
Type | Paracompact uniform honeycomb
|
Schläfli symbol | t0,3{6,3,5} |
Coxeter diagram |
|
Cells | {6,3} {5,3} {}x{6} {}x{5} |
Faces | square {4} pentagon {5} hexagon {6} |
Vertex figure | triangular antiprism
|
Coxeter groups | , [5,3,6] |
Properties | Vertex-transitive |
The runcinated order-5 hexagonal tiling honeycomb, t0,3{6,3,5}, has
Runcitruncated order-5 hexagonal tiling honeycomb
Runcitruncated order-5 hexagonal tiling honeycomb | |
---|---|
Type | Paracompact uniform honeycomb
|
Schläfli symbol | t0,1,3{6,3,5} |
Coxeter diagram |
|
Cells | t{6,3} rr{5,3} {}x{5} {}x{12} |
Faces | triangle {3} square {4} pentagon {5} dodecagon {12} |
Vertex figure | isosceles-trapezoidal pyramid |
Coxeter groups | , [5,3,6] |
Properties | Vertex-transitive |
The runcitruncated order-5 hexagonal tiling honeycomb, t0,1,3{6,3,5}, has truncated hexagonal tiling, rhombicosidodecahedron, pentagonal prism, and dodecagonal prism cells, with an isosceles-trapezoidal pyramid vertex figure.
Runcicantellated order-5 hexagonal tiling honeycomb
The runcicantellated order-5 hexagonal tiling honeycomb is the same as the
Omnitruncated order-5 hexagonal tiling honeycomb
Omnitruncated order-5 hexagonal tiling honeycomb | |
---|---|
Type | Paracompact uniform honeycomb
|
Schläfli symbol | t0,1,2,3{6,3,5} |
Coxeter diagram |
|
Cells | |
Faces | square {4} hexagon {6} decagon {10} dodecagon {12} |
Vertex figure | irregular tetrahedron |
Coxeter groups | , [5,3,6] |
Properties | Vertex-transitive |
The omnitruncated order-5 hexagonal tiling honeycomb, t0,1,2,3{6,3,5}, has
Alternated order-5 hexagonal tiling honeycomb
Alternated order-5 hexagonal tiling honeycomb | |
---|---|
Type | Semiregular honeycomb
|
Schläfli symbol | h{6,3,5} |
Coxeter diagram |
↔ |
Cells | {3[3]} {3,5} |
Faces | triangle {3} |
Vertex figure | truncated icosahedron |
Coxeter groups | , [5,3[3]] |
Properties | Vertex-transitive, edge-transitive, quasiregular
|
The alternated order-5 hexagonal tiling honeycomb, h{6,3,5}, ↔ , has
Cantic order-5 hexagonal tiling honeycomb
Cantic order-5 hexagonal tiling honeycomb | |
---|---|
Type | Paracompact uniform honeycomb
|
Schläfli symbol | h2{6,3,5} |
Coxeter diagram |
↔ |
Cells | h2{6,3} t{3,5} r{5,3} |
Faces | triangle {3} pentagon {5} hexagon {6} |
Vertex figure | triangular prism |
Coxeter groups | , [5,3[3]] |
Properties | Vertex-transitive |
The cantic order-5 hexagonal tiling honeycomb, h2{6,3,5}, ↔ , has trihexagonal tiling, truncated icosahedron, and icosidodecahedron facets, with a triangular prism vertex figure.
Runcic order-5 hexagonal tiling honeycomb
Runcic order-5 hexagonal tiling honeycomb | |
---|---|
Type | Paracompact uniform honeycomb
|
Schläfli symbol | h3{6,3,5} |
Coxeter diagram |
↔ |
Cells | {3[3]} rr{5,3} {5,3} {}x{3} |
Faces | triangle {3} square {4} pentagon {5} |
Vertex figure | triangular cupola |
Coxeter groups | , [5,3[3]] |
Properties | Vertex-transitive |
The runcic order-5 hexagonal tiling honeycomb, h3{6,3,5}, ↔ , has triangular tiling, rhombicosidodecahedron, dodecahedron, and triangular prism facets, with a triangular cupola vertex figure.
Runcicantic order-5 hexagonal tiling honeycomb
Runcicantic order-5 hexagonal tiling honeycomb | |
---|---|
Type | Paracompact uniform honeycomb
|
Schläfli symbol | h2,3{6,3,5} |
Coxeter diagram |
↔ |
Cells | h2{6,3} tr{5,3} t{5,3} {}x{3} |
Faces | triangle {3} square {4} hexagon {6} decagon {10} |
Vertex figure | rectangular pyramid |
Coxeter groups | , [5,3[3]] |
Properties | Vertex-transitive |
The runcicantic order-5 hexagonal tiling honeycomb, h2,3{6,3,5}, ↔ , has trihexagonal tiling, truncated icosidodecahedron, truncated dodecahedron, and triangular prism facets, with a rectangular pyramid vertex figure.
See also
- Convex uniform honeycombs in hyperbolic space
- Regular tessellations of hyperbolic 3-space
- Paracompact uniform honeycombs
References
- ^ Coxeter The Beauty of Geometry, 1999, Chapter 10, Table III
- 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, ISBN 0-486-40919-8 (Chapter 10, Regular Honeycombs in Hyperbolic Space) Table III
- 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