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	${\displaystyle {\overline {BV}}_{3}}$, [4,3,6] ${\displaystyle {\overline {DV}}_{3}}$, [6,31,1] ${\displaystyle {\overline {BP}}_{3}}$, [4,3[3]] ${\displaystyle {\overline {DP}}_{3}}$, [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:

r{p,3,6}

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
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	${\displaystyle {\overline {BV}}_{3}}$, [4,3,6] ${\displaystyle {\overline {BP}}_{3}}$, [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:

The bitruncated order-6 cubic honeycomb is the same as the bitruncated order-4 hexagonal tiling 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	${\displaystyle {\overline {BV}}_{3}}$, [4,3,6] ${\displaystyle {\overline {BP}}_{3}}$, [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
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	${\displaystyle {\overline {BV}}_{3}}$, [4,3,6] ${\displaystyle {\overline {BP}}_{3}}$, [4,3[3]]
Properties	Vertex-transitive

The cantitruncated order-6 cubic honeycomb, tr{4,3,6}, has truncated cuboctahedron, hexagonal tiling, and hexagonal prism facets, with a mirrored sphenoid vertex figure.

The runcinated order-6 cubic honeycomb is the same as the runcinated order-4 hexagonal tiling 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	${\displaystyle {\overline {BV}}_{3}}$, [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.

The runcicantellated order-6 cubic honeycomb is the same as the runcitruncated order-4 hexagonal tiling honeycomb.

The omnitruncated order-6 cubic honeycomb is the same as the omnitruncated order-4 hexagonal tiling honeycomb.

Alternated order-6 cubic honeycomb
Type	Paracompact uniform honeycomb Semiregular honeycomb
Schläfli symbol	h{4,3,6}
Coxeter diagram	↔ ↔ ↔ ↔
Cells	{3,3} {3,6}
Faces	triangle {3}
Vertex figure	trihexagonal tiling
Coxeter group	${\displaystyle {\overline {DV}}_{3}}$, [6,31,1] ${\displaystyle {\overline {DP}}_{3}}$, [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 tessellation (or honeycomb). As an alternation, with Schläfli symbol h{4,3,6} and Coxeter-Dynkin diagram or , it can be considered a quasiregular honeycomb, alternating triangular tilings and tetrahedra around each vertex in a trihexagonal tiling vertex figure.

A half-symmetry construction from the form {4,3^[3]} exists, with two alternating types (colors) of triangular tiling cells. This form has Coxeter-Dynkin diagram ↔ . Another lower-symmetry form of index 6, [4,3^*,6], exists with a non-simplex fundamental domain, with Coxeter-Dynkin diagram .

The alternated order-6 cubic honeycomb is part of a series of quasiregular polychora and honeycombs.

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}
	${\displaystyle \left\{3,{3 \atop 3}\right\}}$	${\displaystyle \left\{3,{4 \atop 3}\right\}}$	${\displaystyle \left\{3,{5 \atop 3}\right\}}$	${\displaystyle \left\{3,{6 \atop 3}\right\}}$	${\displaystyle \left\{4,{4 \atop 3}\right\}}$	${\displaystyle \left\{4,{4 \atop 4}\right\}}$
Coxeter diagram	↔	↔	↔	↔	↔	↔
	↔					↔
Image
Vertex figure r{p,3}

It also has 3 related forms: the cantic order-6 cubic honeycomb, h₂{4,3,6}, ; the runcic order-6 cubic honeycomb, h₃{4,3,6}, ; and the runcicantic order-6 cubic honeycomb, h_2,3{4,3,6}, .

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	${\displaystyle {\overline {DV}}_{3}}$, [6,31,1] ${\displaystyle {\overline {DP}}_{3}}$, [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
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	${\displaystyle {\overline {DV}}_{3}}$, [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
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	${\displaystyle {\overline {DV}}_{3}}$, [6,31,1]
Properties	Vertex-transitive

The runcicantic order-6 cubic honeycomb is a uniform compact space-filling tessellation (or honeycomb), with Schläfli symbol h_2,3{4,3,6}. It is composed of truncated hexagonal tiling, truncated trihexagonal tiling, and truncated tetrahedron facets, with a mirrored sphenoid vertex figure.