Order-4 hexagonal tiling honeycomb
In the field of hyperbolic geometry, the order-4 hexagonal tiling honeycomb arises as 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 flat plane in hyperbolic space that approaches a single ideal point at infinity.
Order-4 hexagonal tiling honeycomb | |
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Perspective projection view within Poincaré disk model | |
Type | Hyperbolic regular honeycomb Paracompact uniform honeycomb |
Schläfli symbols | {6,3,4} {6,31,1} t0,1{(3,6)2} |
Coxeter diagrams | |
Cells | {6,3} |
Faces | hexagon {6} |
Edge figure | square {4} |
Vertex figure | octahedron |
Dual | Order-6 cubic honeycomb |
Coxeter groups | , [4,3,6] , [6,31,1] , [(6,3)[2]] |
Properties | Regular, quasiregular |
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.
The Schläfli symbol of the order-4 hexagonal tiling honeycomb is {6,3,4}. Since that of the hexagonal tiling is {6,3}, this honeycomb has four such hexagonal tilings meeting at each edge. Since the Schläfli symbol of the octahedron is {3,4}, the vertex figure of this honeycomb is an octahedron. Thus, eight hexagonal tilings meet at each vertex of this honeycomb, and the six edges meeting at each vertex lie along three orthogonal axes.[1]
Images
Perspective projection |
One cell, viewed from outside the Poincare sphere |
The vertices of a t{(3,∞,3)}, |
The honeycomb is analogous to the H2 order-4 apeirogonal tiling, {∞,4}, shown here with one green apeirogon outlined by its horocycle |
Symmetry
The order-4 hexagonal tiling honeycomb has three reflective simplex symmetry constructions.
The half-symmetry uniform construction {6,31,1} has two types (colors) of hexagonal tilings, with Coxeter diagram
An additional two reflective symmetries exist with non-simplectic fundamental domains: [6,3*,4], which is index 6, with Coxeter diagram
The order-4 hexagonal tiling honeycomb contains
Related polytopes and honeycombs
The order-4 hexagonal tiling honeycomb is a regular hyperbolic honeycomb in 3-space, and one of 11 which are paracompact.
11 paracompact regular honeycombs | |||||||||||
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{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 fifteen uniform honeycombs in the [6,3,4] Coxeter group family, including this regular form, and its dual, the order-6 cubic honeycomb.
[6,3,4] family honeycombs | |||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|
{6,3,4} | r{6,3,4} | t{6,3,4} | rr{6,3,4} | t0,3{6,3,4} | tr{6,3,4} | t0,1,3{6,3,4} | t0,1,2,3{6,3,4} | ||||
{4,3,6} | r{4,3,6} | t{4,3,6} | rr{4,3,6} | 2t{4,3,6} | tr{4,3,6} | t0,1,3{4,3,6} | t0,1,2,3{4,3,6} |
The order-4 hexagonal tiling honeycomb has a related alternated honeycomb,
It is a part of sequence of regular honeycombs of the form {6,3,p}, all of which are composed of hexagonal tiling cells:
{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,∞} |
This honeycomb is also related to the 16-cell, cubic honeycomb and order-4 dodecahedral honeycomb, all of which have octahedral vertex figures.
{p,3,4} regular honeycombs | |||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|
Space | S3 | E3 | H3 | ||||||||
Form | Finite | Affine | Compact | Paracompact | Noncompact | ||||||
Name | {3,3,4} |
{4,3,4} |
{5,3,4} |
{6,3,4} |
{7,3,4} |
{8,3,4} |
... {∞,3,4} | ||||
Image | |||||||||||
Cells | {3,3} |
{4,3} |
{5,3} |
{6,3} |
{7,3} |
{8,3} |
{∞,3} |
The aforementioned honeycombs are also quasiregular:
Regular and Quasiregular honeycombs: {p,3,4} and {p,31,1} | |||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|
Space | Euclidean 4-space | Euclidean 3-space | Hyperbolic 3-space | ||||||||
Name | {3,3,4} {3,31,1} = |
{4,3,4} {4,31,1} = |
{5,3,4} {5,31,1} = |
{6,3,4} {6,31,1} = | |||||||
Coxeter diagram |
|||||||||||
Image | |||||||||||
Cells {p,3} |
Rectified order-4 hexagonal tiling honeycomb
Rectified order-4 hexagonal tiling honeycomb | |
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Type | Paracompact uniform honeycomb |
Schläfli symbols | r{6,3,4} or t1{6,3,4} |
Coxeter diagrams | |
Cells | {3,4} r{6,3} |
Faces | triangle {3} hexagon {6} |
Vertex figure | square prism |
Coxeter groups | , [4,3,6] , [4,3[3]] , [6,31,1] , [3[]×[]] |
Properties | Vertex-transitive, edge-transitive |
The rectified order-4 hexagonal tiling honeycomb, t1{6,3,4},
It is similar to the 2D hyperbolic tetraapeirogonal tiling, r{∞,4},
Truncated order-4 hexagonal tiling honeycomb
Truncated order-4 hexagonal tiling honeycomb | |
---|---|
Type | Paracompact uniform honeycomb |
Schläfli symbol | t{6,3,4} or t0,1{6,3,4} |
Coxeter diagram | |
Cells | {3,4} t{6,3} |
Faces | triangle {3} dodecagon {12} |
Vertex figure | square pyramid |
Coxeter groups | , [4,3,6] , [6,31,1] |
Properties | Vertex-transitive |
The truncated order-4 hexagonal tiling honeycomb, t0,1{6,3,4},
It is similar to the 2D hyperbolic truncated order-4 apeirogonal tiling, t{∞,4},
Bitruncated order-4 hexagonal tiling honeycomb
Bitruncated order-4 hexagonal tiling honeycomb | |
---|---|
Type | Paracompact uniform honeycomb |
Schläfli symbol | 2t{6,3,4} or t1,2{6,3,4} |
Coxeter diagram | |
Cells | t{4,3} t{3,6} |
Faces | square {4} hexagon {6} |
Vertex figure | digonal disphenoid |
Coxeter groups | , [4,3,6] , [4,3[3]] , [6,31,1] , [3[]×[]] |
Properties | Vertex-transitive |
The bitruncated order-4 hexagonal tiling honeycomb, t1,2{6,3,4},
Cantellated order-4 hexagonal tiling honeycomb
Cantellated order-4 hexagonal tiling honeycomb | |
---|---|
Type | Paracompact uniform honeycomb |
Schläfli symbol | rr{6,3,4} or t0,2{6,3,4} |
Coxeter diagram | |
Cells | r{3,4} {}x{4} rr{6,3} |
Faces | triangle {3} square {4} hexagon {6} |
Vertex figure | wedge |
Coxeter groups | , [4,3,6] , [6,31,1] |
Properties | Vertex-transitive |
The cantellated order-4 hexagonal tiling honeycomb, t0,2{6,3,4},
Cantitruncated order-4 hexagonal tiling honeycomb
Cantitruncated order-4 hexagonal tiling honeycomb | |
---|---|
Type | Paracompact uniform honeycomb |
Schläfli symbol | tr{6,3,4} or t0,1,2{6,3,4} |
Coxeter diagram | |
Cells | t{3,4} {}x{4} tr{6,3} |
Faces | square {4} hexagon {6} dodecagon {12} |
Vertex figure | mirrored sphenoid |
Coxeter groups | , [4,3,6] , [6,31,1] |
Properties | Vertex-transitive |
The cantitruncated order-4 hexagonal tiling honeycomb, t0,1,2{6,3,4},
Runcinated order-4 hexagonal tiling honeycomb
Runcinated order-4 hexagonal tiling honeycomb | |
---|---|
Type | Paracompact uniform honeycomb |
Schläfli symbol | t0,3{6,3,4} |
Coxeter diagram | |
Cells | {4,3} {}x{4} {6,3} {}x{6} |
Faces | square {4} hexagon {6} |
Vertex figure | irregular triangular antiprism |
Coxeter groups | , [4,3,6] |
Properties | Vertex-transitive |
The runcinated order-4 hexagonal tiling honeycomb, t0,3{6,3,4},
It contains the 2D hyperbolic rhombitetrahexagonal tiling, rr{4,6},
Runcitruncated order-4 hexagonal tiling honeycomb
Runcitruncated order-4 hexagonal tiling honeycomb | |
---|---|
Type | Paracompact uniform honeycomb |
Schläfli symbol | t0,1,3{6,3,4} |
Coxeter diagram | |
Cells | rr{3,4} {}x{4} {}x{12} t{6,3} |
Faces | triangle {3} square {4} dodecagon {12} |
Vertex figure | isosceles-trapezoidal pyramid |
Coxeter groups | , [4,3,6] |
Properties | Vertex-transitive |
The runcitruncated order-4 hexagonal tiling honeycomb, t0,1,3{6,3,4},
Runcicantellated order-4 hexagonal tiling honeycomb
The runcicantellated order-4 hexagonal tiling honeycomb is the same as the runcitruncated order-6 cubic honeycomb.
Omnitruncated order-4 hexagonal tiling honeycomb
Omnitruncated order-4 hexagonal tiling honeycomb | |
---|---|
Type | Paracompact uniform honeycomb |
Schläfli symbol | t0,1,2,3{6,3,4} |
Coxeter diagram | |
Cells | tr{4,3} tr{6,3} {}x{12} {}x{8} |
Faces | square {4} hexagon {6} octagon {8} dodecagon {12} |
Vertex figure | irregular tetrahedron |
Coxeter groups | , [4,3,6] |
Properties | Vertex-transitive |
The omnitruncated order-4 hexagonal tiling honeycomb, t0,1,2,3{6,3,4},
Alternated order-4 hexagonal tiling honeycomb
Alternated order-4 hexagonal tiling honeycomb | |
---|---|
Type | Paracompact uniform honeycomb Semiregular honeycomb |
Schläfli symbols | h{6,3,4} |
Coxeter diagrams | |
Cells | {3[3]} {3,4} |
Faces | triangle {3} |
Vertex figure | truncated octahedron |
Coxeter groups | , [4,3[3]] |
Properties | Vertex-transitive, edge-transitive, quasiregular |
The alternated order-4 hexagonal tiling honeycomb,
Cantic order-4 hexagonal tiling honeycomb
Cantic order-4 hexagonal tiling honeycomb | |
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Type | Paracompact uniform honeycomb |
Schläfli symbols | h2{6,3,4} |
Coxeter diagrams | |
Cells | h2{6,3} t{3,4} r{3,4} |
Faces | triangle {3} square {4} hexagon {6} |
Vertex figure | wedge |
Coxeter groups | , [4,3[3]] |
Properties | Vertex-transitive |
The cantic order-4 hexagonal tiling honeycomb,
Runcic order-4 hexagonal tiling honeycomb
Runcic order-4 hexagonal tiling honeycomb | |
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Type | Paracompact uniform honeycomb |
Schläfli symbols | h3{6,3,4} |
Coxeter diagrams | |
Cells | {3[3]} rr{3,4} {4,3} {}x{3} |
Faces | triangle {3} square {4} |
Vertex figure | triangular cupola |
Coxeter groups | , [4,3[3]] |
Properties | Vertex-transitive |
The runcic order-4 hexagonal tiling honeycomb,
Runcicantic order-4 hexagonal tiling honeycomb
Runcicantic order-4 hexagonal tiling honeycomb | |
---|---|
Type | Paracompact uniform honeycomb |
Schläfli symbols | h2,3{6,3,4} |
Coxeter diagrams | |
Cells | h2{6,3} tr{3,4} t{4,3} {}x{3} |
Faces | triangle {3} square {4} hexagon {6} octagon {8} |
Vertex figure | rectangular pyramid |
Coxeter groups | , [4,3[3]] |
Properties | Vertex-transitive |
The runcicantic order-4 hexagonal tiling honeycomb,
Quarter order-4 hexagonal tiling honeycomb
Quarter order-4 hexagonal tiling honeycomb | |
---|---|
Type | Paracompact uniform honeycomb |
Schläfli symbol | q{6,3,4} |
Coxeter diagram | |
Cells | {3[3]} {3,3} t{3,3} h2{6,3} |
Faces | triangle {3} hexagon {6} |
Vertex figure | triangular cupola |
Coxeter groups | , [3[]x[]] |
Properties | Vertex-transitive |
The quarter order-4 hexagonal tiling honeycomb, q{6,3,4},
See also
References
- 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, (2018) Chapter 13: Hyperbolic Coxeter groups