Icosahedral honeycomb

Rectified icosahedral honeycomb
Type	Uniform honeycombs in hyperbolic space
Schläfli symbol	r{3,5,3} or t1{3,5,3}
Coxeter diagram
Cells	r{3,5} {5,3}
Faces	triangle {3} pentagon {5}
Vertex figure	triangular prism
Coxeter group	${\displaystyle {\overline {J}}_{3}}$, [3,5,3]
Properties	Vertex-transitive, edge-transitive

The rectified icosahedral honeycomb, t₁{3,5,3}, , has alternating dodecahedron and icosidodecahedron cells, with a triangular prism vertex figure:

Perspective projections from center of Poincaré disk model

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

Truncated icosahedral honeycomb
Type	Uniform honeycombs in hyperbolic space
Schläfli symbol	t{3,5,3} or t0,1{3,5,3}
Coxeter diagram
Cells	t{3,5} {5,3}
Faces	pentagon {5} hexagon {6}
Vertex figure	triangular pyramid
Coxeter group	${\displaystyle {\overline {J}}_{3}}$, [3,5,3]
Properties	Vertex-transitive

The truncated icosahedral honeycomb, t_0,1{3,5,3}, , has alternating dodecahedron and truncated icosahedron cells, with a triangular pyramid vertex figure.

Bitruncated icosahedral honeycomb
Type	Uniform honeycombs in hyperbolic space
Schläfli symbol	2t{3,5,3} or t1,2{3,5,3}
Coxeter diagram
Cells	t{5,3}
Faces	triangle {3} decagon {10}
Vertex figure	tetragonal disphenoid
Coxeter group	${\displaystyle 2\times {\overline {J}}_{3}}$, [[3,5,3]]
Properties	Vertex-transitive, edge-transitive, cell-transitive

The bitruncated icosahedral honeycomb, t_1,2{3,5,3}, , has truncated dodecahedron cells with a tetragonal disphenoid vertex figure.

Cantellated icosahedral honeycomb
Type	Uniform honeycombs in hyperbolic space
Schläfli symbol	rr{3,5,3} or t0,2{3,5,3}
Coxeter diagram
Cells	rr{3,5} r{5,3} {}x{3}
Faces	triangle {3} square {4} pentagon {5}
Vertex figure	wedge
Coxeter group	${\displaystyle {\overline {J}}_{3}}$, [3,5,3]
Properties	Vertex-transitive

The cantellated icosahedral honeycomb, t_0,2{3,5,3}, , has rhombicosidodecahedron, icosidodecahedron, and triangular prism cells, with a wedge vertex figure.

Cantitruncated icosahedral honeycomb
Type	Uniform honeycombs in hyperbolic space
Schläfli symbol	tr{3,5,3} or t0,1,2{3,5,3}
Coxeter diagram
Cells	tr{3,5} t{5,3} {}x{3}
Faces	triangle {3} square {4} hexagon {6} decagon {10}
Vertex figure	mirrored sphenoid
Coxeter group	${\displaystyle {\overline {J}}_{3}}$, [3,5,3]
Properties	Vertex-transitive

The cantitruncated icosahedral honeycomb, t_0,1,2{3,5,3}, , has truncated icosidodecahedron, truncated dodecahedron, and triangular prism cells, with a mirrored sphenoid vertex figure.

Runcinated icosahedral honeycomb
Type	Uniform honeycombs in hyperbolic space
Schläfli symbol	t0,3{3,5,3}
Coxeter diagram
Cells	{3,5} {}×{3}
Faces	triangle {3} square {4}
Vertex figure	pentagonal antiprism
Coxeter group	${\displaystyle 2\times {\overline {J}}_{3}}$, [[3,5,3]]
Properties	Vertex-transitive, edge-transitive

The runcinated icosahedral honeycomb, t_0,3{3,5,3}, , has icosahedron and triangular prism cells, with a pentagonal antiprism vertex figure.

Viewed from center of triangular prism

Runcitruncated icosahedral honeycomb
Type	Uniform honeycombs in hyperbolic space
Schläfli symbol	t0,1,3{3,5,3}
Coxeter diagram
Cells	t{3,5} rr{3,5} {}×{3} {}×{6}
Faces	triangle {3} square {4} pentagon {5} hexagon {6}
Vertex figure	isosceles-trapezoidal pyramid
Coxeter group	${\displaystyle {\overline {J}}_{3}}$, [3,5,3]
Properties	Vertex-transitive

The runcitruncated icosahedral honeycomb, t_0,1,3{3,5,3}, , has truncated icosahedron, rhombicosidodecahedron, hexagonal prism, and triangular prism cells, with an isosceles-trapezoidal pyramid vertex figure.

The runcicantellated icosahedral honeycomb is equivalent to the runcitruncated icosahedral honeycomb.

Viewed from center of triangular prism

Omnitruncated icosahedral honeycomb
Type	Uniform honeycombs in hyperbolic space
Schläfli symbol	t0,1,2,3{3,5,3}
Coxeter diagram
Cells	tr{3,5} {}×{6}
Faces	square {4} hexagon {6} dodecagon {10}
Vertex figure	phyllic disphenoid
Coxeter group	${\displaystyle 2\times {\overline {J}}_{3}}$, [[3,5,3]]
Properties	Vertex-transitive

The omnitruncated icosahedral honeycomb, t_0,1,2,3{3,5,3}, , has truncated icosidodecahedron and hexagonal prism cells, with a phyllic disphenoid vertex figure.

Centered on hexagonal prism

Omnisnub icosahedral honeycomb
Type	Uniform honeycombs in hyperbolic space
Schläfli symbol	h(t0,1,2,3{3,5,3})
Coxeter diagram
Cells	sr{3,5} s{2,3} irr. {3,3}
Faces	triangle {3} pentagon {5}
Vertex figure
Coxeter group	[[3,5,3]]+
Properties	Vertex-transitive

The omnisnub icosahedral honeycomb, h(t_0,1,2,3{3,5,3}), , has snub dodecahedron, octahedron, and tetrahedron cells, with an irregular vertex figure. It is vertex-transitive, but cannot be made with uniform cells.

Partially diminished icosahedral honeycomb Parabidiminished icosahedral honeycomb
Type	Uniform honeycombs
Schläfli symbol	pd{3,5,3}
Coxeter diagram	-
Cells	{5,3} s{2,5}
Faces	triangle {3} pentagon {5}
Vertex figure	tetrahedrally diminished dodecahedron
Coxeter group	1/5[3,5,3]+
Properties	Vertex-transitive

The partially diminished icosahedral honeycomb or parabidiminished icosahedral honeycomb, pd{3,5,3}, is a non-Wythoffian uniform honeycomb with dodecahedron and pentagonal antiprism cells, with a tetrahedrally diminished dodecahedron vertex figure. The icosahedral cells of the {3,5,3} are diminished at opposite vertices (parabidiminished), leaving a pentagonal antiprism (parabidiminished icosahedron) core, and creating new dodecahedron cells above and below.[1][2]