Order-5 cubic honeycomb explained

bgcolor=#e7dcc3 colspan=2	Order-5 cubic honeycomb
bgcolor=#ffffff align=center colspan=2	Poincaré disk models
Type	Hyperbolic regular honeycomb Uniform hyperbolic honeycomb
Schläfli symbol
Coxeter diagram
Cells	(cube)
Faces	(square)
Edge figure	(pentagon)
Vertex figure	icosahedron
Coxeter group
Dual	Order-4 dodecahedral honeycomb
Properties	Regular

In hyperbolic geometry, the order-5 cubic honeycomb is one of four compact regular space-filling tessellations (or honeycombs) in hyperbolic 3-space. With Schläfli symbol it has five cubes around each edge, and 20 cubes around each vertex. It is dual with the order-4 dodecahedral honeycomb.

Symmetry

It has a radical subgroup symmetry construction with dodecahedral fundamental domains: Coxeter notation: [4,(3,5)^*], index 120.

Related polytopes and honeycombs

The order-5 cubic honeycomb has a related alternated honeycomb, ↔, with icosahedron and tetrahedron cells.

The honeycomb is also one of four regular compact honeycombs in 3D hyperbolic space:

There are fifteen uniform honeycombs in the [5,3,4] Coxeter group family, including the order-5 cubic honeycomb as the regular form:

The order-5 cubic honeycomb is in a sequence of regular polychora and honeycombs with icosahedral vertex figures.

It is also in a sequence of regular polychora and honeycombs with cubic cells. The first polytope in the sequence is the tesseract, and the second is the Euclidean cubic honeycomb.

Rectified order-5 cubic honeycomb

bgcolor=#e7dcc3 colspan=2	Rectified order-5 cubic honeycomb	-	bgcolor=#ffffff align=center colspan=2	-->
Type	Uniform honeycombs in hyperbolic space
Schläfli symbol	r or 2r 2r
width=120 bgcolor=#e7dcc3	Coxeter diagram	↔
Cells
Faces
Vertex figure	pentagonal prism
Coxeter group	\overline{BH} _{3} , [4,3,5] \overline{DH} _{3} , [5,3<sup>1,1</sup>]
Properties	Vertex-transitive, edge-transitive

The rectified order-5 cubic honeycomb,, has alternating icosahedron and cuboctahedron cells, with a pentagonal prism vertex figure.

Related honeycomb

There are four rectified compact regular honeycombs:

Truncated order-5 cubic honeycomb

bgcolor=#e7dcc3 colspan=2	Truncated order-5 cubic honeycomb	-	bgcolor=#ffffff align=center colspan=2	-->
Type	Uniform honeycombs in hyperbolic space
Schläfli symbol	t
width=120 bgcolor=#e7dcc3	Coxeter diagram
Cells
Faces
Vertex figure	pentagonal pyramid
Coxeter group	\overline{BH} _{3} , [4,3,5]
Properties	Vertex-transitive

The truncated order-5 cubic honeycomb,, has truncated cube and icosahedron cells, with a pentagonal pyramid vertex figure.

It can be seen as analogous to the 2D hyperbolic truncated order-5 square tiling, t, with truncated square and pentagonal faces:

It is similar to the Euclidean (order-4) truncated cubic honeycomb, t, which has octahedral cells at the truncated vertices.

Related honeycombs

Bitruncated order-5 cubic honeycomb

The bitruncated order-5 cubic honeycomb is the same as the bitruncated order-4 dodecahedral honeycomb.

Cantellated order-5 cubic honeycomb

bgcolor=#e7dcc3 colspan=2	Cantellated order-5 cubic honeycomb	-	bgcolor=#ffffff align=center colspan=2	-->
Type	Uniform honeycombs in hyperbolic space
Schläfli symbol	rr
width=120 bgcolor=#e7dcc3	Coxeter diagram
Cells
Faces
Vertex figure	wedge
Coxeter group	\overline{BH} _{3} , [4,3,5]
Properties	Vertex-transitive

The cantellated order-5 cubic honeycomb,, has rhombicuboctahedron, icosidodecahedron, and pentagonal prism cells, with a wedge vertex figure.

Related honeycombs

It is similar to the Euclidean (order-4) cantellated cubic honeycomb, rr:

Cantitruncated order-5 cubic honeycomb

bgcolor=#e7dcc3 colspan=2	Cantitruncated order-5 cubic honeycomb	-	bgcolor=#ffffff align=center colspan=2	-->
Type	Uniform honeycombs in hyperbolic space
Schläfli symbol	tr
width=120 bgcolor=#e7dcc3	Coxeter diagram
Cells
Faces
Vertex figure	mirrored sphenoid
Coxeter group	\overline{BH} _{3} , [4,3,5]
Properties	Vertex-transitive

The cantitruncated order-5 cubic honeycomb,, has truncated cuboctahedron, truncated icosahedron, and pentagonal prism cells, with a mirrored sphenoid vertex figure.

Related honeycombs

It is similar to the Euclidean (order-4) cantitruncated cubic honeycomb, tr:

Runcinated order-5 cubic honeycomb

bgcolor=#e7dcc3 colspan=2	Runcinated order-5 cubic honeycomb	-	bgcolor=#ffffff align=center colspan=2	-->
Type	Uniform honeycombs in hyperbolic space Semiregular honeycomb
Schläfli symbol	t0,3
width=120 bgcolor=#e7dcc3	Coxeter diagram
Cells
Faces
Vertex figure	irregular triangular antiprism
Coxeter group	\overline{BH} _{3} , [4,3,5]
Properties	Vertex-transitive

The runcinated order-5 cubic honeycomb or runcinated order-4 dodecahedral honeycomb, has cube, dodecahedron, and pentagonal prism cells, with an irregular triangular antiprism vertex figure.

It is analogous to the 2D hyperbolic rhombitetrapentagonal tiling, rr, with square and pentagonal faces:

Related honeycombs

It is similar to the Euclidean (order-4) runcinated cubic honeycomb, t_0,3:

Runcitruncated order-5 cubic honeycomb

bgcolor=#e7dcc3 colspan=2	Runctruncated order-5 cubic honeycomb Runcicantellated order-4 dodecahedral honeycomb	-	bgcolor=#ffffff align=center colspan=2	-->
Type	Uniform honeycombs in hyperbolic space
Schläfli symbol	t0,1,3
width=120 bgcolor=#e7dcc3	Coxeter diagram
Cells
Faces
Vertex figure	isosceles-trapezoidal pyramid
Coxeter group	\overline{BH} _{3} , [4,3,5]
Properties	Vertex-transitive

The runcitruncated order-5 cubic honeycomb or runcicantellated order-4 dodecahedral honeycomb,, has truncated cube, rhombicosidodecahedron, pentagonal prism, and octagonal prism cells, with an isosceles-trapezoidal pyramid vertex figure.

Related honeycombs

It is similar to the Euclidean (order-4) runcitruncated cubic honeycomb, t_0,1,3:

Runcicantellated order-5 cubic honeycomb

The runcicantellated order-5 cubic honeycomb is the same as the runcitruncated order-4 dodecahedral honeycomb.

Omnitruncated order-5 cubic honeycomb

bgcolor=#e7dcc3 colspan=2	Omnitruncated order-5 cubic honeycomb	-	bgcolor=#ffffff align=center colspan=2	-->
Type	Uniform honeycombs in hyperbolic space Semiregular honeycomb
Schläfli symbol	t0,1,2,3
width=120 bgcolor=#e7dcc3	Coxeter diagram
Cells
Faces
Vertex figure	irregular tetrahedron
Coxeter group	\overline{BH} _{3} , [4,3,5]
Properties	Vertex-transitive

The omnitruncated order-5 cubic honeycomb or omnitruncated order-4 dodecahedral honeycomb,, has truncated icosidodecahedron, truncated cuboctahedron, decagonal prism, and octagonal prism cells, with an irregular tetrahedral vertex figure.

Related honeycombs

It is similar to the Euclidean (order-4) omnitruncated cubic honeycomb, t_0,1,2,3:

Alternated order-5 cubic honeycomb

bgcolor=#e7dcc3 colspan=2	Alternated order-5 cubic honeycomb	-	bgcolor=#ffffff align=center colspan=2	-->
Type	Uniform honeycombs in hyperbolic space
Schläfli symbol	h
Coxeter diagram	↔
Cells
Faces
Vertex figure	icosidodecahedron
Coxeter group	\overline{DH} _{3} , [5,3<sup>1,1</sup>]
Properties	Vertex-transitive, edge-transitive, quasiregular

In 3-dimensional hyperbolic geometry, the alternated order-5 cubic honeycomb is a uniform compact space-filling tessellation (or honeycomb). With Schläfli symbol h, it can be considered a quasiregular honeycomb, alternating icosahedra and tetrahedra around each vertex in an icosidodecahedron vertex figure.

Related honeycombs

It has 3 related forms: the cantic order-5 cubic honeycomb,, the runcic order-5 cubic honeycomb,, and the runcicantic order-5 cubic honeycomb, .

Cantic order-5 cubic honeycomb

bgcolor=#e7dcc3 colspan=2	Cantic order-5 cubic honeycomb
Type	Uniform honeycombs in hyperbolic space
Schläfli symbol	h2
Coxeter diagram	↔
Cells
Faces
Vertex figure	rectangular pyramid
Coxeter group	\overline{DH} _{3} , [5,3<sup>1,1</sup>]
Properties	Vertex-transitive

The cantic order-5 cubic honeycomb is a uniform compact space-filling tessellation (or honeycomb), with Schläfli symbol h₂. It has icosidodecahedron, truncated icosahedron, and truncated tetrahedron cells, with a rectangular pyramid vertex figure.

Runcic order-5 cubic honeycomb

bgcolor=#e7dcc3 colspan=2	Runcic order-5 cubic honeycomb
Type	Uniform honeycombs in hyperbolic space
Schläfli symbol	h3
Coxeter diagram	↔
Cells
Faces
Vertex figure	triangular frustum
Coxeter group	\overline{DH} _{3} , [5,3<sup>1,1</sup>]
Properties	Vertex-transitive

The runcic order-5 cubic honeycomb is a uniform compact space-filling tessellation (or honeycomb), with Schläfli symbol h₃. It has dodecahedron, rhombicosidodecahedron, and tetrahedron cells, with a triangular frustum vertex figure.

Runcicantic order-5 cubic honeycomb

bgcolor=#e7dcc3 colspan=2	Runcicantic order-5 cubic honeycomb
Type	Uniform honeycombs in hyperbolic space
Schläfli symbol	h2,3
Coxeter diagram	↔
Cells
Faces
Vertex figure	irregular tetrahedron
Coxeter group	\overline{DH} _{3} , [5,3<sup>1,1</sup>]
Properties	Vertex-transitive

The runcicantic order-5 cubic honeycomb is a uniform compact space-filling tessellation (or honeycomb), with Schläfli symbol h_2,3. It has truncated dodecahedron, truncated icosidodecahedron, and truncated tetrahedron cells, with an irregular tetrahedron vertex figure.

See also

• Convex uniform honeycombs in hyperbolic space
• Regular tessellations of hyperbolic 3-space

References

• Coxeter, Regular Polytopes, 3rd. ed., Dover Publications, 1973. . (Tables I and II: Regular polytopes and honeycombs, pp. 294–296)
• Coxeter, The Beauty of Geometry: Twelve Essays, Dover Publications, 1999 (Chapter 10: Regular honeycombs in hyperbolic space, Summary tables II, III, IV, V, p212-213)
• 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