Order-5 hexagonal tiling honeycomb

Rectified order-5 hexagonal tiling honeycomb
Type	Paracompact uniform honeycomb
Schläfli symbols	r{6,3,5} or t1{6,3,5}
Coxeter diagrams	↔
Cells	{3,5} r{6,3} or h2{6,3}
Faces	triangle {3} hexagon {6}
Vertex figure	pentagonal prism
Coxeter groups	${\displaystyle {{\overline {HV}}_{3}}}$, [5,3,6] ${\displaystyle {{\overline {HP}}_{3}}}$, [5,3[3]]
Properties	Vertex-transitive, edge-transitive

The rectified order-5 hexagonal tiling honeycomb, t₁{6,3,5}, has icosahedron and trihexagonal tiling facets, with a pentagonal prism vertex figure.

It is similar to the 2D hyperbolic infinite-order square tiling, r{∞,5} with pentagon and apeirogonal faces. All vertices are on the ideal surface.

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 hexagonal tiling honeycomb
Type	Paracompact uniform honeycomb
Schläfli symbol	t{6,3,5} or t0,1{6,3,5}
Coxeter diagram
Cells	{3,5} t{6,3}
Faces	triangle {3} dodecagon {12}
Vertex figure	pentagonal pyramid
Coxeter groups	${\displaystyle {\overline {HV}}_{3}}$, [5,3,6]
Properties	Vertex-transitive

The truncated order-5 hexagonal tiling honeycomb, t_0,1{6,3,5}, has icosahedron and truncated hexagonal tiling facets, with a pentagonal pyramid vertex figure.

Bitruncated order-5 hexagonal tiling honeycomb
Type	Paracompact uniform honeycomb
Schläfli symbol	2t{6,3,5} or t1,2{6,3,5}
Coxeter diagram	↔
Cells	t{3,6} t{3,5}
Faces	pentagon {5} hexagon {6}
Vertex figure	digonal disphenoid
Coxeter groups	${\displaystyle {\overline {HV}}_{3}}$, [5,3,6] ${\displaystyle {\overline {HP}}_{3}}$, [5,3[3]]
Properties	Vertex-transitive

The bitruncated order-5 hexagonal tiling honeycomb, t_1,2{6,3,5}, has hexagonal tiling and truncated icosahedron facets, with a digonal disphenoid vertex figure.

Cantellated order-5 hexagonal tiling honeycomb
Type	Paracompact uniform honeycomb
Schläfli symbol	rr{6,3,5} or t0,2{6,3,5}
Coxeter diagram
Cells	r{3,5} rr{6,3} {}x{5}
Faces	triangle {3} square {4} pentagon {5} hexagon {6}
Vertex figure	wedge
Coxeter groups	${\displaystyle {\overline {HV}}_{3}}$, [5,3,6]
Properties	Vertex-transitive

The cantellated order-5 hexagonal tiling honeycomb, t_0,2{6,3,5}, has icosidodecahedron, rhombitrihexagonal tiling, and pentagonal prism facets, with a wedge vertex figure.

Cantitruncated order-5 hexagonal tiling honeycomb
Type	Paracompact uniform honeycomb
Schläfli symbol	tr{6,3,5} or t0,1,2{6,3,5}
Coxeter diagram
Cells	t{3,5} tr{6,3} {}x{5}
Faces	square {4} pentagon {5} hexagon {6} dodecagon {12}
Vertex figure	mirrored sphenoid
Coxeter groups	${\displaystyle {\overline {HV}}_{3}}$, [5,3,6]
Properties	Vertex-transitive

The cantitruncated order-5 hexagonal tiling honeycomb, t_0,1,2{6,3,5}, has truncated icosahedron, truncated trihexagonal tiling, and pentagonal prism facets, with a mirrored sphenoid vertex figure.

Runcinated order-5 hexagonal tiling honeycomb
Type	Paracompact uniform honeycomb
Schläfli symbol	t0,3{6,3,5}
Coxeter diagram
Cells	{6,3} {5,3} {}x{6} {}x{5}
Faces	square {4} pentagon {5} hexagon {6}
Vertex figure	irregular triangular antiprism
Coxeter groups	${\displaystyle {\overline {HV}}_{3}}$, [5,3,6]
Properties	Vertex-transitive

The runcinated order-5 hexagonal tiling honeycomb, t_0,3{6,3,5}, has dodecahedron, hexagonal tiling, pentagonal prism, and hexagonal prism facets, with an irregular triangular antiprism vertex figure.

Runcitruncated order-5 hexagonal tiling honeycomb
Type	Paracompact uniform honeycomb
Schläfli symbol	t0,1,3{6,3,5}
Coxeter diagram
Cells	t{6,3} rr{5,3} {}x{5} {}x{12}
Faces	triangle {3} square {4} pentagon {5} dodecagon {12}
Vertex figure	isosceles-trapezoidal pyramid
Coxeter groups	${\displaystyle {\overline {HV}}_{3}}$, [5,3,6]
Properties	Vertex-transitive

The runcitruncated order-5 hexagonal tiling honeycomb, t_0,1,3{6,3,5}, has truncated hexagonal tiling, rhombicosidodecahedron, pentagonal prism, and dodecagonal prism cells, with an isosceles-trapezoidal pyramid vertex figure.

The runcicantellated order-5 hexagonal tiling honeycomb is the same as the runcitruncated order-6 dodecahedral honeycomb.

Omnitruncated order-5 hexagonal tiling honeycomb
Type	Paracompact uniform honeycomb
Schläfli symbol	t0,1,2,3{6,3,5}
Coxeter diagram
Cells	tr{6,3} tr{5,3} {}x{10} {}x{12}
Faces	square {4} hexagon {6} decagon {10} dodecagon {12}
Vertex figure	irregular tetrahedron
Coxeter groups	${\displaystyle {\overline {HV}}_{3}}$, [5,3,6]
Properties	Vertex-transitive

The omnitruncated order-5 hexagonal tiling honeycomb, t_0,1,2,3{6,3,5}, has truncated trihexagonal tiling, truncated icosidodecahedron, decagonal prism, and dodecagonal prism facets, with an irregular tetrahedral vertex figure.

Alternated order-5 hexagonal tiling honeycomb
Type	Paracompact uniform honeycomb Semiregular honeycomb
Schläfli symbol	h{6,3,5}
Coxeter diagram	↔
Cells	{3[3]} {3,5}
Faces	triangle {3}
Vertex figure	truncated icosahedron
Coxeter groups	${\displaystyle {\overline {HP}}_{3}}$, [5,3[3]]
Properties	Vertex-transitive, edge-transitive, quasiregular

The alternated order-5 hexagonal tiling honeycomb, h{6,3,5}, ↔ , has triangular tiling and icosahedron facets, with a truncated icosahedron vertex figure. It is a quasiregular honeycomb.

Cantic order-5 hexagonal tiling honeycomb
Type	Paracompact uniform honeycomb
Schläfli symbol	h2{6,3,5}
Coxeter diagram	↔
Cells	h2{6,3} t{3,5} r{5,3}
Faces	triangle {3} pentagon {5} hexagon {6}
Vertex figure	triangular prism
Coxeter groups	${\displaystyle {\overline {HP}}_{3}}$, [5,3[3]]
Properties	Vertex-transitive

The cantic order-5 hexagonal tiling honeycomb, h₂{6,3,5}, ↔ , has trihexagonal tiling, truncated icosahedron, and icosidodecahedron facets, with a triangular prism vertex figure.

Runcic order-5 hexagonal tiling honeycomb
Type	Paracompact uniform honeycomb
Schläfli symbol	h3{6,3,5}
Coxeter diagram	↔
Cells	{3[3]} rr{5,3} {5,3} {}x{3}
Faces	triangle {3} square {4} pentagon {5}
Vertex figure	triangular cupola
Coxeter groups	${\displaystyle {\overline {HP}}_{3}}$, [5,3[3]]
Properties	Vertex-transitive

The runcic order-5 hexagonal tiling honeycomb, h₃{6,3,5}, ↔ , has triangular tiling, rhombicosidodecahedron, dodecahedron, and triangular prism facets, with a triangular cupola vertex figure.

Runcicantic order-5 hexagonal tiling honeycomb
Type	Paracompact uniform honeycomb
Schläfli symbol	h2,3{6,3,5}
Coxeter diagram	↔
Cells	h2{6,3} tr{5,3} t{5,3} {}x{3}
Faces	triangle {3} square {4} hexagon {6} decagon {10}
Vertex figure	rectangular pyramid
Coxeter groups	${\displaystyle {\overline {HP}}_{3}}$, [5,3[3]]
Properties	Vertex-transitive

The runcicantic order-5 hexagonal tiling honeycomb, h_2,3{6,3,5}, ↔ , has trihexagonal tiling, truncated icosidodecahedron, truncated dodecahedron, and triangular prism facets, with a rectangular pyramid vertex figure.