Icosahedral honeycomb
The icosahedral honeycomb is one of four compact regular spacefilling tessellations (or honeycombs) in hyperbolic 3space. With Schläfli symbol {3,5,3}, there are three icosahedra, {3,5}, around each edge, and 12 icosahedra around each vertex, in a regular dodecahedral, {5,3}, vertex figure.
Icosahedral honeycomb  

Poincaré disk model  
Type  regular hyperbolic honeycomb 
Schläfli symbol  {3,5,3} 
Coxeter diagram  
Cells  {3,5} 
Faces  triangle {3} 
Vertex figure  dodecahedron 
Dual  Selfdual 
Coxeter group  J_{3}, [3,5,3] 
Properties  Regular 
A geometric honeycomb is a spacefilling of polyhedral or higherdimensional cells, so that there are no gaps. It is an example of the more general mathematical tiling or tessellation in any number of dimensions.
Honeycombs are usually constructed in ordinary Euclidean ("flat") space, like the convex uniform honeycombs. They may also be constructed in nonEuclidean spaces, such as hyperbolic uniform honeycombs. Any finite uniform polytope can be projected to its circumsphere to form a uniform honeycomb in spherical space.
Description
The dihedral angle of a Euclidean icosahedron is 138.2°, so it is impossible to fit three icosahedra around an edge in Euclidean 3space. However, in hyperbolic space, properly scaled icosahedra can have dihedral angles exactly 120 degrees, so three of these fit around an edge.
Related regular honeycombs
There are four regular compact honeycombs in 3D hyperbolic space:
{5,3,4} 
{4,3,5} 
{3,5,3} 
{5,3,5} 
Related regular polytopes and honeycombs
It a part of a sequence of regular polychora and honeycombs {3,p,3}:
{3,p,3} polytopes  

Space  S^{3}  H^{3}  
Form  Finite  Compact  Paracompact  Noncompact  
{3,p,3}  {3,3,3}  {3,4,3}  {3,5,3}  {3,6,3}  {3,7,3}  {3,8,3}  ... {3,∞,3}  
Image  
Cells  {3,3} 
{3,4} 
{3,5} 
{3,6} 
{3,7} 
{3,8} 
{3,∞}  
Vertex figure 
{3,3} 
{4,3} 
{5,3} 
{6,3} 
{7,3} 
{8,3} 
{∞,3} 
It a part of a sequence of regular polychora and honeycombs {p,5,p}:
{p,5,p} regular honeycombs  

Space  H^{3}  
Form  Paracompact  Noncompact  
Name  {3,5,3}  {4,5,4}  {5,5,5}  {6,5,6}  {7,5,7}  {8,5,8}  ...{∞,5,∞}  
Image  
Cells {p,5} 
{3,5} 
{4,5} 
{5,5} 
{6,5} 
{7,5} 
{8,5} 
{∞,5}  
Vertex figure {5,p} 
{5,3} 
{5,4} 
{5,5} 
{5,6} 
{5,7} 
{5,8} 
{5,∞} 
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}{3,5,3},
{3,5,3} 
t_{1}{3,5,3} 
t_{0,1}{3,5,3} 
t_{0,2}{3,5,3} 
t_{0,3}{3,5,3} 

t_{1,2}{3,5,3} 
t_{0,1,2}{3,5,3} 
t_{0,1,3}{3,5,3} 
t_{0,1,2,3}{3,5,3}  
Rectified icosahedral honeycomb
Rectified icosahedral honeycomb  

Type  Uniform honeycombs in hyperbolic space 
Schläfli symbol  r{3,5,3} or t_{1}{3,5,3} 
Coxeter diagram  
Cells  r{3,5} {5,3} 
Faces  triangle {3} Pentagon {5} 
Vertex figure  Triangular prism 
Coxeter group  J_{3}, [3,5,3] 
Properties  Vertextransitive, edgetransitive 
The rectified icosahedral honeycomb, t_{1}{3,5,3},
Perspective projections from center of Poincaré disk model
Related honeycomb
There are four rectified compact regular honeycombs:
Image  

Symbols  r{5,3,4} 
r{4,3,5} 
r{3,5,3} 
r{5,3,5} 
Vertex figure 
Truncated icosahedral honeycomb
Truncated icosahedral honeycomb  

Type  Uniform honeycombs in hyperbolic space 
Schläfli symbol  t{3,5,3} or t_{0,1}{3,5,3} 
Coxeter diagram  
Cells  t{3,5} {5,3} 
Faces  triangle {3} Pentagon {5} 
Vertex figure  triangular pyramid 
Coxeter group  J_{3}, [3,5,3] 
Properties  Vertextransitive 
The truncated icosahedral honeycomb, t_{0,1}{3,5,3},
Related honeycombs
Image  

Symbols  t{5,3,4} 
t{4,3,5} 
t{3,5,3} 
t{5,3,5} 
Vertex figure 
Bitruncated icosahedral honeycomb
Bitruncated icosahedral honeycomb  

Type  Uniform honeycombs in hyperbolic space 
Schläfli symbol  2t{3,5,3} or t_{1,2}{3,5,3} 
Coxeter diagram  
Cells  t{5,3} 
Faces  Triangle {3} Decagon {10} 
Vertex figure  disphenoid 
Coxeter group  J_{3}×2, [[3,5,3]] 
Properties  Vertextransitive, edgetransitive, celltransitive 
The bitruncated icosahedral honeycomb, t_{1,2}{3,5,3},
Related honeycombs
Image  

Symbols  2t{4,3,5} 
2t{3,5,3} 
2t{5,3,5} 
Vertex figure 
Cantellated icosahedral honeycomb
Cantellated icosahedral honeycomb  

Type  Uniform honeycombs in hyperbolic space 
Schläfli symbol  rr{3,5,3} or t_{0,2}{3,5,3} 
Coxeter diagram  
Cells  rr{3,5} r{5,3} 
Faces  triangle {3} Square {4} Pentagon {5} 
Vertex figure  triangular prism 
Coxeter group  J_{3}, [3,5,3] 
Properties  Vertextransitive 
The cantellated icosahedral honeycomb, t_{0,2}{3,5,3},
Related honeycombs
Four cantellated regular compact honeycombs in H^{3}  


Cantitruncated icosahedral honeycomb
Cantitruncated icosahedral honeycomb  

Type  Uniform honeycombs in hyperbolic space 
Schläfli symbol  tr{3,5,3} or t_{0,1,2}{3,5,3} 
Coxeter diagram  
Cells  tr{3,5} r{5,3} {}x{3} {}x{6} 
Faces  Triangle {3} Square {4} Pentagon {5} Hexagon {6} 
Vertex figure  Mirrored sphenoid 
Coxeter group  J_{3}, [3,5,3] 
Properties  Vertextransitive 
The cantitruncated icosahedral honeycomb, t_{0,1,2}{3,5,3},
Related honeycombs
Image  

Symbols  tr{5,3,4} 
tr{4,3,5} 
tr{3,5,3} 
tr{5,3,5} 
Vertex figure 
Runcinated icosahedral honeycomb
Runcinated icosahedral honeycomb  

Type  Uniform honeycombs in hyperbolic space 
Schläfli symbol  t_{0,3}{3,5,3} 
Coxeter diagram  
Cells  {3,5} {}×{3} 
Faces  Triangle {3} Square {4} 
Vertex figure  pentagonal antiprism 
Coxeter group  J_{3}×2, [[3,5,3]] 
Properties  Vertextransitive 
The runcinated icosahedral honeycomb, t_{0,3}{3,5,3},
 Viewed from center of triangular prism
Related honeycombs
Image  

Symbols  t_{0,3}{4,3,5} 
t_{0,3}{3,5,3} 
t_{0,3}{5,3,5} 
Vertex figure 
Runcitruncated icosahedral honeycomb
Runcitruncated icosahedral honeycomb  

Type  Uniform honeycombs in hyperbolic space 
Schläfli symbol  t_{0,1,3}{3,5,3} 
Coxeter diagram  
Cells  r{3,5} rr{3,5} {}×{3} {}×{6} 
Faces  Triangle {3} Square {4} Pentagon {5} Hexagon {6} 
Vertex figure  square pyramid 
Coxeter group  J_{3}, [3,5,3] 
Properties  Vertextransitive 
The runcitruncated icosahedral honeycomb, t_{0,1,3}{3,5,3},
 Viewed from center of triangular prism
Related honeycombs
Four runcitruncated regular compact honeycombs in H^{3}  


Omnitruncated icosahedral honeycomb
Omnitruncated icosahedral honeycomb  

Type  Uniform honeycombs in hyperbolic space 
Schläfli symbol  t_{0,1,2,3}{3,5,3} 
Coxeter diagram  
Cells  tr{3,5} {}×{6} 
Faces  Square {4} Hexagon {6} Dodecagon {10} 
Vertex figure  Phyllic disphenoid 
Coxeter group  J_{3}×2, [[3,5,3]] 
Properties  Vertextransitive 
The omnitruncated icosahedral honeycomb, t_{0,1,2,3}{3,5,3},
 Centered on hexagonal prism
Related honeycombs
Three omnitruncated regular compact honeycombs in H^{3}  


Omnisnub icosahedral honeycomb
Omnisnub icosahedral honeycomb  

Type  Uniform honeycombs in hyperbolic space 
Schläfli symbol  h(t_{0,1,2,3}{3,5,3}) 
Coxeter diagram  
Cells  sr{3,5} s{2,3} irr. {3,3} 
Faces  Square {4} Pentagon {5} 
Vertex figure  
Coxeter group  J_{3}×2, [[3,5,3]]^{+} 
Properties  Vertextransitive 
The omnisnub icosahedral honeycomb, h(t_{0,1,2,3}{3,5,3}),
Partially diminished icosahedral honeycomb
Partially diminished icosahedral honeycomb Parabidiminished icosahedral honeycomb  

Type  Uniform honeycombs 
Schläfli symbol  pd{3,5,3} 
Coxeter diagram   
Cells  {5,3} s{2,10} 
Faces  Triangle {3} Pentagon {5} 
Vertex figure  tetrahedrally diminished dodecahedron 
Coxeter group  ^{1}/_{5}[3,5,3]^{+} 
Properties  Vertextransitive 
The partially diminished icosahedral honeycomb or parabidiminished icosahedral honeycomb, pd{3,5,3}, is a nonwythoffian 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]
See also
 Seifert–Weber space
 List of regular polytopes
 Convex uniform honeycombs in hyperbolic space
 11cell  An abstract regular polychoron which shares the {3,5,3} Schläfli symbol.
References
 Wendy Y. Krieger, Walls and bridges: The view from six dimensions, Symmetry: Culture and Science Volume 16, Number 2, pages 171–192 (2005)
 http://www.bendwavy.org/klitzing/incmats/pt353.htm
 Coxeter, Regular Polytopes, 3rd. ed., Dover Publications, 1973. ISBN 0486614808. (Tables I and II: Regular polytopes and honeycombs, pp. 294–296)
 Coxeter, The Beauty of Geometry: Twelve Essays, Dover Publications, 1999 ISBN 0486409198 (Chapter 10: Regular honeycombs in hyperbolic space, Summary tables II,III,IV,V, p212213)
 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
 Klitzing, Richard. "Hyperbolic H3 honeycombs hyperbolic order 3 icosahedral tesselation".