Order-6 cubic honeycomb
Order-6 cubic honeycomb | |
---|---|
Perspective projection view
within Poincaré disk model | |
Type | Paracompact uniform honeycomb
|
Schläfli symbol | {4,3,6} {4,3[3]} |
Coxeter diagram |
↔ ↔ |
Cells |
{4,3} |
Faces | square {4}
|
Edge figure | hexagon {6} |
Vertex figure | triangular tiling |
Coxeter group | , [4,3,6] , [4,3[3]] |
Dual |
Order-4 hexagonal tiling honeycomb |
Properties | Regular, quasiregular
|
The order-6 cubic honeycomb is a paracompact
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
Images
One cell viewed outside of the Poincaré sphere model |
The order-6 cubic honeycomb is analogous to the 2D hyperbolic infinite-order square tiling, {4,∞} with square faces. All vertices are on the ideal surface. |
Symmetry
A half-symmetry construction of the order-6 cubic honeycomb exists as {4,3[3]}, with two alternating types (colors) of cubic cells. This construction has
Another lower-symmetry construction, [4,3*,6], of index 6, exists with a non-simplex fundamental domain, with
This honeycomb contains that tile 2-hypercycle surfaces, similar to the paracompact order-3 apeirogonal tiling, :
Related polytopes and honeycombs
The order-6 cubic 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} |
It has a related alternation honeycomb, represented by ↔ . This alternated form has hexagonal tiling and tetrahedron cells.
There are
[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-6 cubic honeycomb is part of a sequence of
{4,3,p} regular honeycombs | |||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|
Space | S3 | E3 | H3 | ||||||||
Form | Finite | Affine | Compact | Paracompact | Noncompact | ||||||
Name |
{4,3,3} |
{4,3,4} |
{4,3,5} |
{4,3,6} |
{4,3,7} |
{4,3,8} |
... {4,3,∞} | ||||
Image | |||||||||||
Vertex figure |
{3,3} |
{3,4} |
{3,5} |
{3,6} |
{3,7} |
{3,8} |
{3,∞} |
It is also part of a sequence of honeycombs with
Form | Paracompact | Noncompact | |||||
---|---|---|---|---|---|---|---|
Name | {3,3,6} | {4,3,6} | {5,3,6} | {6,3,6} | {7,3,6}
|
{8,3,6}
|
... {∞,3,6}
|
Image | |||||||
Cells | {3,3} |
{4,3} |
{5,3} |
{6,3} |
{7,3} |
{8,3} |
{∞,3} |
Rectified order-6 cubic honeycomb
Rectified order-6 cubic honeycomb | |
---|---|
Type | Paracompact uniform honeycomb
|
Schläfli symbols | r{4,3,6} or t1{4,3,6} |
Coxeter diagrams |
↔ ↔ ↔ |
Cells | r{3,4} {3,6} |
Faces | triangle {3} square {4} |
Vertex figure | hexagonal prism |
Coxeter groups | , [4,3,6] , [6,31,1] , [4,3[3]] , [3[]×[]] |
Properties | Vertex-transitive, edge-transitive |
The rectified order-6 cubic honeycomb, r{4,3,6}, has cuboctahedral and triangular tiling facets, with a hexagonal prism vertex figure.
It is similar to the 2D hyperbolic tetraapeirogonal tiling, r{4,∞}, alternating apeirogonal and square faces:
Space | H3 | ||||||
---|---|---|---|---|---|---|---|
Form | Paracompact | Noncompact | |||||
Name | r{3,3,6} |
r{4,3,6} |
r{5,3,6} |
r{6,3,6} |
r{7,3,6} |
... r{∞,3,6} | |
Image | |||||||
Cells {3,6} |
r{3,3} |
r{4,3} |
r{5,3} |
r{6,3} |
r{7,3} |
r{∞,3} |
Truncated order-6 cubic honeycomb
Truncated order-6 cubic honeycomb | |
---|---|
Type | Paracompact uniform honeycomb
|
Schläfli symbols | t{4,3,6} or t0,1{4,3,6} |
Coxeter diagrams |
↔ |
Cells | t{4,3} {3,6} |
Faces | triangle {3} octagon {8} |
Vertex figure | hexagonal pyramid |
Coxeter groups | , [4,3,6] , [4,3[3]] |
Properties | Vertex-transitive |
The truncated order-6 cubic honeycomb, t{4,3,6}, has truncated cube and triangular tiling facets, with a hexagonal pyramid vertex figure.
It is similar to the 2D hyperbolic truncated infinite-order square tiling, t{4,∞}, with apeirogonal and octagonal (truncated square) faces:
Bitruncated order-6 cubic honeycomb
The bitruncated order-6 cubic honeycomb is the same as the
Cantellated order-6 cubic honeycomb
Cantellated order-6 cubic honeycomb | |
---|---|
Type | Paracompact uniform honeycomb
|
Schläfli symbols | rr{4,3,6} or t0,2{4,3,6} |
Coxeter diagrams |
↔ |
Cells | rr{4,3} r{3,6} {}x{6} |
Faces | triangle {3} square {4} hexagon {6} |
Vertex figure | wedge |
Coxeter groups | , [4,3,6] , [4,3[3]] |
Properties | Vertex-transitive |
The cantellated order-6 cubic honeycomb, rr{4,3,6}, has rhombicuboctahedron, trihexagonal tiling, and hexagonal prism facets, with a wedge vertex figure.
Cantitruncated order-6 cubic honeycomb
Cantitruncated order-6 cubic honeycomb | |
---|---|
Type | Paracompact uniform honeycomb
|
Schläfli symbols | tr{4,3,6} or t0,1,2{4,3,6} |
Coxeter diagrams |
↔ |
Cells | tr{4,3} t{3,6} {}x{6} |
Faces | square {4} hexagon {6} octagon {8} |
Vertex figure | mirrored sphenoid
|
Coxeter groups | , [4,3,6] , [4,3[3]] |
Properties | Vertex-transitive |
The cantitruncated order-6 cubic honeycomb, tr{4,3,6}, has
Runcinated order-6 cubic honeycomb
The runcinated order-6 cubic honeycomb is the same as the
Runcitruncated order-6 cubic honeycomb
Cantellated order-6 cubic honeycomb | |
---|---|
Type | Paracompact uniform honeycomb
|
Schläfli symbols | t0,1,3{4,3,6} |
Coxeter diagrams |
|
Cells | t{4,3} rr{3,6} {}x{6} {}x{8} |
Faces | triangle {3} square {4} hexagon {6} octagon {8} |
Vertex figure | isosceles-trapezoidal pyramid |
Coxeter groups | , [4,3,6] |
Properties | Vertex-transitive |
The runcitruncated order-6 cubic honeycomb, rr{4,3,6}, has truncated cube, rhombitrihexagonal tiling, hexagonal prism, and octagonal prism facets, with an isosceles-trapezoidal pyramid vertex figure.
Runcicantellated order-6 cubic honeycomb
The runcicantellated order-6 cubic honeycomb is the same as the
Omnitruncated order-6 cubic honeycomb
The omnitruncated order-6 cubic honeycomb is the same as the
Alternated order-6 cubic honeycomb
Alternated order-6 cubic honeycomb | |
---|---|
Type | Semiregular honeycomb
|
Schläfli symbol | h{4,3,6} |
Coxeter diagram |
↔ ↔ ↔ ↔ |
Cells | {3,3} {3,6} |
Faces | triangle {3} |
Vertex figure | trihexagonal tiling |
Coxeter group | , [6,31,1] , [3[]x[]] |
Properties | Vertex-transitive, edge-transitive, quasiregular
|
In three-dimensional hyperbolic geometry, the alternated order-6 hexagonal tiling honeycomb is a uniform compact space-filling
Symmetry
A half-symmetry construction from the form {4,3[3]} exists, with two alternating types (colors) of triangular tiling cells. This form has
Related honeycombs
The alternated order-6 cubic honeycomb is part of a series of
Quasiregular polychora and honeycombs: h{4,p,q} | |||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|
Space | Finite | Affine | Compact | Paracompact | |||||||
Schläfli symbol |
h{4,3,3} | h{4,3,4}
|
h{4,3,5}
|
h{4,3,6}
|
h{4,4,3}
|
h{4,4,4} | |||||
Coxeter
diagram |
↔ | ↔ | ↔ | ↔ | ↔ | ↔ | |||||
↔ | ↔ | ||||||||||
Image | |||||||||||
Vertex figure r{p,3} |
It also has 3 related forms: the
Cantic order-6 cubic honeycomb
Cantic order-6 cubic honeycomb | |
---|---|
Type | Paracompact uniform honeycomb
|
Schläfli symbol | h2{4,3,6} |
Coxeter diagram |
↔ ↔ ↔ |
Cells | t{3,3} r{6,3} t{3,6} |
Faces | triangle {3} hexagon {6} |
Vertex figure | rectangular pyramid |
Coxeter group | , [6,31,1] , [3[]x[]] |
Properties | Vertex-transitive |
The cantic order-6 cubic honeycomb is a uniform compact space-filling tessellation (or honeycomb) with Schläfli symbol h2{4,3,6}. It is composed of truncated tetrahedron, trihexagonal tiling, and hexagonal tiling facets, with a rectangular pyramid vertex figure.
Runcic order-6 cubic honeycomb
Runcic order-6 cubic honeycomb | |
---|---|
Type | Paracompact uniform honeycomb
|
Schläfli symbol | h3{4,3,6} |
Coxeter diagram |
↔ |
Cells | {3,3} {6,3} rr{6,3} |
Faces | triangle {3} square {4} hexagon {6} |
Vertex figure | triangular cupola |
Coxeter group | , [6,31,1] |
Properties | Vertex-transitive |
The runcic order-6 cubic honeycomb is a uniform compact space-filling tessellation (or honeycomb) with Schläfli symbol h3{4,3,6}. It is composed of tetrahedron, hexagonal tiling, and rhombitrihexagonal tiling facets, with a triangular cupola vertex figure.
Runcicantic order-6 cubic honeycomb
Runcicantic order-6 cubic honeycomb | |
---|---|
Type | Paracompact uniform honeycomb
|
Schläfli symbol | h2,3{4,3,6} |
Coxeter diagram |
↔ |
Cells | t{6,3} tr{6,3} t{3,3} |
Faces | triangle {3} square {4} hexagon {6} dodecagon {12} |
Vertex figure | mirrored sphenoid
|
Coxeter group | , [6,31,1] |
Properties | Vertex-transitive |
The runcicantic order-6 cubic honeycomb is a uniform compact space-filling
See also
- Convex uniform honeycombs in hyperbolic space
- Regular tessellations of hyperbolic 3-space
- Paracompact uniform honeycombs
References
- 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