Order-6 hexagonal tiling honeycomb

Rectified order-6 hexagonal tiling honeycomb
Type	Paracompact uniform honeycomb
Schläfli symbols	r{6,3,6} or t1{6,3,6}
Coxeter diagrams	↔ ↔ ↔ ↔
Cells	{3,6} r{6,3}
Faces	triangle {3} hexagon {6}
Vertex figure	hexagonal prism
Coxeter groups	${\displaystyle {\overline {Z}}_{3}}$, [6,3,6] ${\displaystyle {\overline {VP}}_{3}}$, [6,3[3]] ${\displaystyle {\overline {PP}}_{3}}$, [3[3,3]]
Properties	Vertex-transitive, edge-transitive

The rectified order-6 hexagonal tiling honeycomb, t₁{6,3,6}, has triangular tiling and trihexagonal tiling facets, with a hexagonal prism vertex figure.

it can also be seen as a quarter order-6 hexagonal tiling honeycomb, q{6,3,6}, ↔ .

It is analogous to 2D hyperbolic order-4 apeirogonal tiling, r{∞,∞} with infinite apeirogonal faces, and with all vertices on the ideal surface.

The order-6 hexagonal tiling honeycomb is part of a series of honeycombs with hexagonal prism vertex figures:

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}

It is also part of a matrix of 3-dimensional quarter honeycombs: q{2p,4,2q}

Euclidean/hyperbolic(paracompact/noncompact) quarter honeycombs q{p,3,q}
p \ q	4	6	8	... ∞
4	q{4,3,4} ↔ ↔	q{4,3,6} ↔ ↔	q{4,3,8} ↔	q{4,3,∞} ↔
6	q{6,3,4} ↔ ↔	q{6,3,6} ↔	q{6,3,8} ↔	q{6,3,∞} ↔
8	q{8,3,4} ↔	q{8,3,6} ↔	q{8,3,8} ↔	q{8,3,∞} ↔
... ∞	q{∞,3,4} ↔	q{∞,3,6} ↔	q{∞,3,8} ↔	q{∞,3,∞} ↔

Truncated order-6 hexagonal tiling honeycomb
Type	Paracompact uniform honeycomb
Schläfli symbol	t{6,3,6} or t0,1{6,3,6}
Coxeter diagram	↔
Cells	{3,6} t{6,3}
Faces	triangle {3} dodecagon {12}
Vertex figure	hexagonal pyramid
Coxeter groups	${\displaystyle {\overline {Z}}_{3}}$, [6,3,6] ${\displaystyle {\overline {VP}}_{3}}$, [6,3[3]]
Properties	Vertex-transitive

The truncated order-6 hexagonal tiling honeycomb, t_0,1{6,3,6}, has triangular tiling and truncated hexagonal tiling facets, with a hexagonal pyramid vertex figure.[2]

Bitruncated order-6 hexagonal tiling honeycomb
Type	Paracompact uniform honeycomb
Schläfli symbol	bt{6,3,6} or t1,2{6,3,6}
Coxeter diagram	↔
Cells	t{3,6}
Faces	hexagon {6}
Vertex figure	tetrahedron
Coxeter groups	${\displaystyle 2\times {\overline {Z}}_{3}}$, [[6,3,6]] ${\displaystyle {\overline {VP}}_{3}}$, [6,3[3]] ${\displaystyle {\overline {V}}_{3}}$, [3,3,6]
Properties	Regular

The bitruncated order-6 hexagonal tiling honeycomb is a lower symmetry construction of the regular hexagonal tiling honeycomb, ↔ . It contains hexagonal tiling facets, with a tetrahedron vertex figure.

Cantellated order-6 hexagonal tiling honeycomb
Type	Paracompact uniform honeycomb
Schläfli symbol	rr{6,3,6} or t0,2{6,3,6}
Coxeter diagram	↔
Cells	r{3,6} rr{6,3} {}x{6}
Faces	triangle {3} square {4} hexagon {6}
Vertex figure	wedge
Coxeter groups	${\displaystyle {\overline {Z}}_{3}}$, [6,3,6] ${\displaystyle {\overline {VP}}_{3}}$, [6,3[3]]
Properties	Vertex-transitive

The cantellated order-6 hexagonal tiling honeycomb, t_0,2{6,3,6}, has trihexagonal tiling, rhombitrihexagonal tiling, and hexagonal prism cells, with a wedge vertex figure.

Cantitruncated order-6 hexagonal tiling honeycomb
Type	Paracompact uniform honeycomb
Schläfli symbol	tr{6,3,6} or t0,1,2{6,3,6}
Coxeter diagram	↔
Cells	tr{3,6} t{3,6} {}x{6}
Faces	triangle {3} square {4} hexagon {6} dodecagon {12}
Vertex figure	mirrored sphenoid
Coxeter groups	${\displaystyle {\overline {Z}}_{3}}$, [6,3,6] ${\displaystyle {\overline {VP}}_{3}}$, [6,3[3]]
Properties	Vertex-transitive

The cantitruncated order-6 hexagonal tiling honeycomb, t_0,1,2{6,3,6}, has hexagonal tiling, truncated trihexagonal tiling, and hexagonal prism cells, with a mirrored sphenoid vertex figure.

Runcinated order-6 hexagonal tiling honeycomb
Type	Paracompact uniform honeycomb
Schläfli symbol	t0,3{6,3,6}
Coxeter diagram	↔
Cells	{6,3} {}×{6}
Faces	triangle {3} square {4} hexagon {6}
Vertex figure	triangular antiprism
Coxeter groups	${\displaystyle 2\times {\overline {Z}}_{3}}$, [[6,3,6]]
Properties	Vertex-transitive, edge-transitive

The runcinated order-6 hexagonal tiling honeycomb, t_0,3{6,3,6}, has hexagonal tiling and hexagonal prism cells, with a triangular antiprism vertex figure.

It is analogous to the 2D hyperbolic rhombihexahexagonal tiling, rr{6,6}, with square and hexagonal faces:

Runcitruncated order-6 hexagonal tiling honeycomb
Type	Paracompact uniform honeycomb
Schläfli symbol	t0,1,3{6,3,6}
Coxeter diagram
Cells	t{6,3} rr{6,3} {}x{6} {}x{12}
Faces	triangle {3} square {4} hexagon {6} dodecagon {12}
Vertex figure	isosceles-trapezoidal pyramid
Coxeter groups	${\displaystyle {\overline {Z}}_{3}}$, [6,3,6]
Properties	Vertex-transitive

The runcitruncated order-6 hexagonal tiling honeycomb, t_0,1,3{6,3,6}, has truncated hexagonal tiling, rhombitrihexagonal tiling, hexagonal prism, and dodecagonal prism cells, with an isosceles-trapezoidal pyramid vertex figure.

Omnitruncated order-6 hexagonal tiling honeycomb
Type	Paracompact uniform honeycomb
Schläfli symbol	t0,1,2,3{6,3,6}
Coxeter diagram
Cells	tr{6,3} {}x{12}
Faces	square {4} hexagon {6} dodecagon {12}
Vertex figure	phyllic disphenoid
Coxeter groups	${\displaystyle 2\times {\overline {Z}}_{3}}$, [[6,3,6]]
Properties	Vertex-transitive

The omnitruncated order-6 hexagonal tiling honeycomb, t_0,1,2,3{6,3,6}, has truncated trihexagonal tiling and dodecagonal prism cells, with a phyllic disphenoid vertex figure.

Alternated order-6 hexagonal tiling honeycomb
Type	Paracompact uniform honeycomb
Schläfli symbols	h{6,3,6}
Coxeter diagrams	↔
Cells	{3,6} {3[3]}
Faces	triangle {3}
Vertex figure	hexagonal tiling
Coxeter groups	${\displaystyle {\overline {VP}}_{3}}$, [6,3[3]]
Properties	Regular, quasiregular

The alternated order-6 hexagonal tiling honeycomb is a lower-symmetry construction of the regular triangular tiling honeycomb, ↔ . It contains triangular tiling facets in a hexagonal tiling vertex figure.

Cantic order-6 hexagonal tiling honeycomb
Type	Paracompact uniform honeycomb
Schläfli symbols	h2{6,3,6}
Coxeter diagrams	↔
Cells	t{3,6} r{6,3} h2{6,3}
Faces	triangle {3} hexagon {6}
Vertex figure	triangular prism
Coxeter groups	${\displaystyle {\overline {VP}}_{3}}$, [6,3[3]]
Properties	Vertex-transitive, edge-transitive

The cantic order-6 hexagonal tiling honeycomb is a lower-symmetry construction of the rectified triangular tiling honeycomb, ↔ , with trihexagonal tiling and hexagonal tiling facets in a triangular prism vertex figure.

Runcic order-6 hexagonal tiling honeycomb
Type	Paracompact uniform honeycomb
Schläfli symbols	h3{6,3,6}
Coxeter diagrams	↔
Cells	rr{3,6} {6,3} {3[3]} {3}x{}
Faces	triangle {3} square {4} hexagon {6}
Vertex figure	triangular cupola
Coxeter groups	${\displaystyle {\overline {VP}}_{3}}$, [6,3[3]]
Properties	Vertex-transitive

The runcic hexagonal tiling honeycomb, h₃{6,3,6}, , or , has hexagonal tiling, rhombitrihexagonal tiling, triangular tiling, and triangular prism facets, with a triangular cupola vertex figure.

Runcicantic order-6 hexagonal tiling honeycomb
Type	Paracompact uniform honeycomb
Schläfli symbols	h2,3{6,3,6}
Coxeter diagrams	↔
Cells	tr{6,3} t{6,3} h2{6,3} {}x{3}
Faces	triangle {3} square {4} hexagon {6} dodecagon {12}
Vertex figure	rectangular pyramid
Coxeter groups	${\displaystyle {\overline {VP}}_{3}}$, [6,3[3]]
Properties	Vertex-transitive

The runcicantic order-6 hexagonal tiling honeycomb, h_2,3{6,3,6}, , or , contains truncated trihexagonal tiling, truncated hexagonal tiling, trihexagonal tiling, and triangular prism facets, with a rectangular pyramid vertex figure.