Order-5 cubic honeycomb

Rectified order-5 cubic honeycomb
Type	Uniform honeycombs in hyperbolic space
Schläfli symbol	r{4,3,5} or 2r{5,3,4} 2r{5,31,1}
Coxeter diagram	↔
Cells	r{4,3} {3,5}
Faces	triangle {3} square {4}
Vertex figure	pentagonal prism
Coxeter group	${\displaystyle {\overline {BH}}_{3}}$, [4,3,5] ${\displaystyle {\overline {DH}}_{3}}$, [5,31,1]
Properties	Vertex-transitive, edge-transitive

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

It can be seen as analogous to the 2D hyperbolic tetrapentagonal tiling, r{4,5} with square and pentagonal faces

There are four rectified compact regular honeycombs:

Four rectified regular compact honeycombs in H³

Image
Symbols	r{5,3,4}	r{4,3,5}	r{3,5,3}	r{5,3,5}
Vertex figure

r{p,3,5}

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 cubic honeycomb
Type	Uniform honeycombs in hyperbolic space
Schläfli symbol	t{4,3,5}
Coxeter diagram
Cells	t{4,3} {3,5}
Faces	triangle {3} octagon {8}
Vertex figure	pentagonal pyramid
Coxeter group	${\displaystyle {\overline {BH}}_{3}}$, [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{4,5}, with truncated square and pentagonal faces:

It is similar to the Euclidean (order-4) truncated cubic honeycomb, t{4,3,4}, which has octahedral cells at the truncated vertices.

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

Cantellated order-5 cubic honeycomb
Type	Uniform honeycombs in hyperbolic space
Schläfli symbol	rr{4,3,5}
Coxeter diagram
Cells	rr{4,3} r{3,5} {}x{5}
Faces	triangle {3} square {4} pentagon {5}
Vertex figure	wedge
Coxeter group	${\displaystyle {\overline {BH}}_{3}}$, [4,3,5]
Properties	Vertex-transitive

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

It is similar to the Euclidean (order-4) cantellated cubic honeycomb, rr{4,3,4}:

Four cantellated regular compact honeycombs in H3
Image Symbols rr{5,3,4} rr{4,3,5} rr{3,5,3} rr{5,3,5} Vertex figure

Cantitruncated order-5 cubic honeycomb
Type	Uniform honeycombs in hyperbolic space
Schläfli symbol	tr{4,3,5}
Coxeter diagram
Cells	tr{4,3} t{3,5} {}x{5}
Faces	square {4} pentagon {5} hexagon {6} octagon {8}
Vertex figure	mirrored sphenoid
Coxeter group	${\displaystyle {\overline {BH}}_{3}}$, [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.

It is similar to the Euclidean (order-4) cantitruncated cubic honeycomb, tr{4,3,4}:

Four cantitruncated regular compact honeycombs in H³

Image
Symbols	tr{5,3,4}	tr{4,3,5}	tr{3,5,3}	tr{5,3,5}
Vertex figure

Runcinated order-5 cubic honeycomb
Type	Uniform honeycombs in hyperbolic space Semiregular honeycomb
Schläfli symbol	t0,3{4,3,5}
Coxeter diagram
Cells	{4,3} {5,3} {}x{5}
Faces	square {4} pentagon {5}
Vertex figure	irregular triangular antiprism
Coxeter group	${\displaystyle {\overline {BH}}_{3}}$, [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{4,5}, with square and pentagonal faces:

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

Three runcinated regular compact honeycombs in H³

Image
Symbols	t0,3{4,3,5}	t0,3{3,5,3}	t0,3{5,3,5}
Vertex figure

Runctruncated order-5 cubic honeycomb Runcicantellated order-4 dodecahedral honeycomb
Type	Uniform honeycombs in hyperbolic space
Schläfli symbol	t0,1,3{4,3,5}
Coxeter diagram
Cells	t{4,3} rr{5,3} {}x{5} {}x{8}
Faces	triangle {3} square {4} pentagon {5} octagon {8}
Vertex figure	isosceles-trapezoidal pyramid
Coxeter group	${\displaystyle {\overline {BH}}_{3}}$, [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.

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

Four runcitruncated regular compact honeycombs in H3
Image Symbols t0,1,3{5,3,4} t0,1,3{4,3,5} t0,1,3{3,5,3} t0,1,3{5,3,5} Vertex figure

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

Omnitruncated order-5 cubic honeycomb
Type	Uniform honeycombs in hyperbolic space Semiregular honeycomb
Schläfli symbol	t0,1,2,3{4,3,5}
Coxeter diagram
Cells	tr{5,3} tr{4,3} {10}x{} {8}x{}
Faces	square {4} hexagon {6} octagon {8} decagon {10}
Vertex figure	irregular tetrahedron
Coxeter group	${\displaystyle {\overline {BH}}_{3}}$, [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.

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

Three omnitruncated regular compact honeycombs in H3
Image Symbols t0,1,2,3{4,3,5} t0,1,2,3{3,5,3} t0,1,2,3{5,3,5} Vertex figure

Alternated order-5 cubic honeycomb
Type	Uniform honeycombs in hyperbolic space
Schläfli symbol	h{4,3,5}
Coxeter diagram	↔
Cells	{3,3} {3,5}
Faces	triangle {3}
Vertex figure	icosidodecahedron
Coxeter group	${\displaystyle {\overline {DH}}_{3}}$, [5,31,1]
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{4,3,5}, it can be considered a quasiregular honeycomb, alternating icosahedra and tetrahedra around each vertex in an icosidodecahedron vertex figure.

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
Type	Uniform honeycombs in hyperbolic space
Schläfli symbol	h2{4,3,5}
Coxeter diagram	↔
Cells	r{5,3} t{3,5} t{3,3}
Faces	triangle {3} pentagon {5} hexagon {6}
Vertex figure	rectangular pyramid
Coxeter group	${\displaystyle {\overline {DH}}_{3}}$, [5,31,1]
Properties	Vertex-transitive

The cantic order-5 cubic honeycomb is a uniform compact space-filling tessellation (or honeycomb), with Schläfli symbol h₂{4,3,5}. It has icosidodecahedron, truncated icosahedron, and truncated tetrahedron cells, with a rectangular pyramid vertex figure.

Runcic order-5 cubic honeycomb
Type	Uniform honeycombs in hyperbolic space
Schläfli symbol	h3{4,3,5}
Coxeter diagram	↔
Cells	{5,3} rr{5,3} {3,3}
Faces	triangle {3} square {4} pentagon {5}
Vertex figure	triangular frustum
Coxeter group	${\displaystyle {\overline {DH}}_{3}}$, [5,31,1]
Properties	Vertex-transitive

The runcic order-5 cubic honeycomb is a uniform compact space-filling tessellation (or honeycomb), with Schläfli symbol h₃{4,3,5}. It has dodecahedron, rhombicosidodecahedron, and tetrahedron cells, with a triangular frustum vertex figure.

Runcicantic order-5 cubic honeycomb
Type	Uniform honeycombs in hyperbolic space
Schläfli symbol	h2,3{4,3,5}
Coxeter diagram	↔
Cells	t{5,3} tr{5,3} t{3,3}
Faces	triangle {3} square {4} hexagon {6} decagon {10}
Vertex figure	irregular tetrahedron
Coxeter group	${\displaystyle {\overline {DH}}_{3}}$, [5,31,1]
Properties	Vertex-transitive

The runcicantic order-5 cubic honeycomb is a uniform compact space-filling tessellation (or honeycomb), with Schläfli symbol h_2,3{4,3,5}. It has truncated dodecahedron, truncated icosidodecahedron, and truncated tetrahedron cells, with an irregular tetrahedron vertex figure.