Icosahedral honeycomb explained

bgcolor=#e7dcc3 colspan=2	Icosahedral honeycomb
bgcolor=#ffffff align=center colspan=2	Poincaré disk model
Type	Hyperbolic regular honeycomb Uniform hyperbolic honeycomb
Schläfli symbol
Coxeter diagram
Cells	(regular icosahedron)
Faces	(triangle)
Edge figure	(triangle)
Vertex figure	dodecahedron
Dual	Self-dual
Coxeter group
Properties	Regular

In geometry, the icosahedral honeycomb is one of four compact, regular, space-filling tessellations (or honeycombs) in hyperbolic 3-space. With Schläfli symbol there are three icosahedra around each edge, and 12 icosahedra around each vertex, in a regular dodecahedral vertex figure.

Description

The dihedral angle of a regular icosahedron is around 138.2°, so it is impossible to fit three icosahedra around an edge in Euclidean 3-space. However, in hyperbolic space, properly scaled icosahedra can have dihedral angles of exactly 120 degrees, so three of those can fit around an edge.

Related regular honeycombs

There are four regular compact honeycombs in 3D hyperbolic space:

Related regular polytopes and honeycombs

It is a member of a sequence of regular polychora and honeycombs with deltrahedral cells:

It is also a member of a sequence of regular polychora and honeycombs, with vertex figures composed of pentagons:

Uniform honeycombs

There are nine uniform honeycombs in the [3,5,3] Coxeter group family, including this regular form as well as the bitruncated form, t_1,2,, also called truncated dodecahedral honeycomb, each of whose cells are truncated dodecahedra.

Rectified icosahedral honeycomb

bgcolor=#e7dcc3 colspan=2	Rectified icosahedral honeycomb	-	bgcolor=#ffffff align=center colspan=2	-->
Type	Uniform honeycombs in hyperbolic space
width=100 bgcolor=#e7dcc3	Schläfli symbol	r or t1
Coxeter diagram
Cells
Faces
Vertex figure	triangular prism
Coxeter group	\overline{J} _{3} , [3,5,3]
Properties	Vertex-transitive, edge-transitive

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

Perspective projections from center of Poincaré disk model

Related honeycomb

There are four rectified compact regular honeycombs:

Truncated icosahedral honeycomb

bgcolor=#e7dcc3 colspan=2	Truncated icosahedral honeycomb	-	bgcolor=#ffffff align=center colspan=2	-->
Type	Uniform honeycombs in hyperbolic space
width=100 bgcolor=#e7dcc3	Schläfli symbol	t or t0,1
Coxeter diagram
Cells
Faces
Vertex figure	triangular pyramid
Coxeter group	\overline{J} _{3} , [3,5,3]
Properties	Vertex-transitive

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

Related honeycombs

Bitruncated icosahedral honeycomb

bgcolor=#e7dcc3 colspan=2	Bitruncated icosahedral honeycomb	-	bgcolor=#ffffff align=center colspan=2	-->
Type	Uniform honeycombs in hyperbolic space
width=100 bgcolor=#e7dcc3	Schläfli symbol	2t or t1,2
Coxeter diagram
Cells
Faces
Vertex figure	tetragonal disphenoid
Coxeter group	2 x \overline{J} _{3} , [[3,5,3]]
Properties	Vertex-transitive, edge-transitive, cell-transitive

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

Related honeycombs

Cantellated icosahedral honeycomb

bgcolor=#e7dcc3 colspan=2	Cantellated icosahedral honeycomb	-	bgcolor=#ffffff align=center colspan=2	-->
Type	Uniform honeycombs in hyperbolic space
width=100 bgcolor=#e7dcc3	Schläfli symbol	rr or t0,2
Coxeter diagram
Cells
Faces
Vertex figure	wedge
Coxeter group	\overline{J} _{3} , [3,5,3]
Properties	Vertex-transitive

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

Related honeycombs

Cantitruncated icosahedral honeycomb

bgcolor=#e7dcc3 colspan=2	Cantitruncated icosahedral honeycomb	-	bgcolor=#ffffff align=center colspan=2	-->
Type	Uniform honeycombs in hyperbolic space
width=100 bgcolor=#e7dcc3	Schläfli symbol	tr or t0,1,2
Coxeter diagram
Cells
Faces
Vertex figure	mirrored sphenoid
Coxeter group	\overline{J} _{3} , [3,5,3]
Properties	Vertex-transitive

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

Related honeycombs

Runcinated icosahedral honeycomb

bgcolor=#e7dcc3 colspan=2	Runcinated icosahedral honeycomb	-	bgcolor=#ffffff align=center colspan=2	-->
Type	Uniform honeycombs in hyperbolic space
width=100 bgcolor=#e7dcc3	Schläfli symbol	t0,3
Coxeter diagram
Cells
Faces
Vertex figure	pentagonal antiprism
Coxeter group	2 x \overline{J} _{3} , [[3,5,3]]
Properties	Vertex-transitive, edge-transitive

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

Viewed from center of triangular prism

Related honeycombs

Runcitruncated icosahedral honeycomb

bgcolor=#e7dcc3 colspan=2	Runcitruncated icosahedral honeycomb	-	bgcolor=#ffffff align=center colspan=2	-->
Type	Uniform honeycombs in hyperbolic space
width=100 bgcolor=#e7dcc3	Schläfli symbol	t0,1,3
Coxeter diagram
Cells
Faces
Vertex figure	isosceles-trapezoidal pyramid
Coxeter group	\overline{J} _{3} , [3,5,3]
Properties	Vertex-transitive

The runcitruncated icosahedral honeycomb, t_0,1,3,, 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

Related honeycombs

Omnitruncated icosahedral honeycomb

bgcolor=#e7dcc3 colspan=2	Omnitruncated icosahedral honeycomb	-	bgcolor=#ffffff align=center colspan=2	-->
Type	Uniform honeycombs in hyperbolic space
width=100 bgcolor=#e7dcc3	Schläfli symbol	t0,1,2,3
Coxeter diagram
Cells
Faces
Vertex figure	phyllic disphenoid
Coxeter group	2 x \overline{J} _{3} , [[3,5,3]]
Properties	Vertex-transitive

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

Centered on hexagonal prism

Related honeycombs

Omnisnub icosahedral honeycomb

bgcolor=#e7dcc3 colspan=2	Omnisnub icosahedral honeycomb	-	bgcolor=#ffffff align=center colspan=2	-->
Type	Uniform honeycombs in hyperbolic space
width=100 bgcolor=#e7dcc3	Schläfli symbol	h(t0,1,2,3)
Coxeter diagram
Cells
Faces
Vertex figure
Coxeter group	[[3,5,3]]^+
Properties	Vertex-transitive

The omnisnub icosahedral honeycomb, h(t_0,1,2,3),, 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

bgcolor=#e7dcc3 colspan=2	Partially diminished icosahedral honeycomb Parabidiminished icosahedral honeycomb	-	bgcolor=#ffffff align=center colspan=2	-->
Type	Uniform honeycombs
Schläfli symbol	pd
Coxeter diagram	-
Cells
Faces
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, is a non-Wythoffian uniform honeycomb with dodecahedron and pentagonal antiprism cells, with a tetrahedrally diminished dodecahedron vertex figure. The icosahedral cells of the are diminished at opposite vertices (parabidiminished), leaving a pentagonal antiprism (parabidiminished icosahedron) core, and creating new dodecahedron cells above and below.^{[1] [2]}

See also

• Convex uniform honeycombs in hyperbolic space
• Regular tessellations of hyperbolic 3-space
• Seifert–Weber space
• 11-cell - An abstract regular polychoron which shares the Schläfli symbol.

References

• Coxeter, Regular Polytopes, 3rd. ed., Dover Publications, 1973. . (Tables I and II: Regular polytopes and honeycombs, pp. 294–296)
• Coxeter, The Beauty of Geometry: Twelve Essays, Dover Publications, 1999 (Chapter 10: Regular honeycombs in hyperbolic space, Summary tables II, III, IV, V, p212-213)
• Norman Johnson Uniform Polytopes, Manuscript
  • N.W. Johnson: The Theory of Uniform Polytopes and Honeycombs, Ph.D. Dissertation, University of Toronto, 1966
  • N.W. Johnson: Geometries and Transformations, (2018) Chapter 13: Hyperbolic Coxeter groups

Notes and References

1. Wendy Y. Krieger, Walls and bridges: The view from six dimensions, Symmetry: Culture and Science Volume 16, Number 2, pages 171–192 (2005) http://symmetry.hu/content/aus_journal_content_abs_2005_16_2.html
2. Web site: Pd.