Icosahedral honeycomb

Icosahedral honeycomb
H3 353 CC center.png
Poincaré disk model
Type Hyperbolic regular honeycomb
Uniform hyperbolic honeycomb
Schläfli symbol {3,5,3}
Coxeter diagram CDel node 1.pngCDel 3.pngCDel node.pngCDel 5.pngCDel node.pngCDel 3.pngCDel node.png
Cells {3,5} Uniform polyhedron-53-t2.png
Faces triangle {3}
Edge figure triangle {3}
Vertex figure Order-3 icosahedral honeycomb verf.png
dodecahedron
Dual Self-dual
Coxeter group , [3,5,3]
Properties Regular

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

A geometric honeycomb is a space-filling of polyhedral or higher-dimensional 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 non-Euclidean spaces, such as hyperbolic uniform honeycombs. Any finite uniform polytope can be projected to its circumsphere to form a uniform honeycomb in spherical space.

DescriptionEdit

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.

 
Honeycomb seen in perspective outside Poincare's model disk

Related regular honeycombsEdit

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

Four regular compact honeycombs in H3
 
{5,3,4}
 
{4,3,5}
 
{3,5,3}
 
{5,3,5}

Related regular polytopes and honeycombsEdit

It is a member of a sequence of regular polychora and honeycombs {3,p,3} with deltrahedral cells:

It is also a member of a sequence of regular polychora and honeycombs {p,5,p}, with vertex figures composed of pentagons:

Uniform honeycombsEdit

There are nine uniform honeycombs in the [3,5,3] Coxeter group family, including this regular form as well as the bitruncated form, t1,2{3,5,3},        , also called truncated dodecahedral honeycomb, each of whose cells are truncated dodecahedra.

[3,5,3] family honeycombs
{3,5,3}
       
t1{3,5,3}
       
t0,1{3,5,3}
       
t0,2{3,5,3}
       
t0,3{3,5,3}
       
         
t1,2{3,5,3}
       
t0,1,2{3,5,3}
       
t0,1,3{3,5,3}
       
t0,1,2,3{3,5,3}
       
       

Rectified icosahedral honeycombEdit

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  , [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

Related honeycombEdit

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 honeycombEdit

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  , [3,5,3]
Properties Vertex-transitive

The truncated icosahedral honeycomb, t0,1{3,5,3},        , has alternating dodecahedron and truncated icosahedron cells, with a triangular pyramid vertex figure.

 

Related honeycombsEdit

Four truncated regular compact honeycombs in H3
Image        
Symbols t{5,3,4}
       
t{4,3,5}
       
t{3,5,3}
       
t{5,3,5}
       
Vertex
figure
       

Bitruncated icosahedral honeycombEdit

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  , [[3,5,3]]
Properties Vertex-transitive, edge-transitive, cell-transitive

The bitruncated icosahedral honeycomb, t1,2{3,5,3},        , has truncated dodecahedron cells with a tetragonal disphenoid vertex figure.

 

Related honeycombsEdit

Three bitruncated compact honeycombs in H3
Image      
Symbols 2t{4,3,5}
       
2t{3,5,3}
       
2t{5,3,5}
       
Vertex
figure
     

Cantellated icosahedral honeycombEdit

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  , [3,5,3]
Properties Vertex-transitive

The cantellated icosahedral honeycomb, t0,2{3,5,3},        , has rhombicosidodecahedron, icosidodecahedron, and triangular prism cells, with a wedge vertex figure.

 

Related honeycombsEdit

Cantitruncated icosahedral honeycombEdit

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  , [3,5,3]
Properties Vertex-transitive

The cantitruncated icosahedral honeycomb, t0,1,2{3,5,3},        , has truncated icosidodecahedron, truncated dodecahedron, and triangular prism cells, with a mirrored sphenoid vertex figure.

 

Related honeycombsEdit

Four cantitruncated regular compact honeycombs in H3
Image        
Symbols tr{5,3,4}
       
tr{4,3,5}
       
tr{3,5,3}
       
tr{5,3,5}
       
Vertex
figure
       

Runcinated icosahedral honeycombEdit

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  , [[3,5,3]]
Properties Vertex-transitive, edge-transitive

The runcinated icosahedral honeycomb, t0,3{3,5,3},        , has icosahedron and triangular prism cells, with a pentagonal antiprism vertex figure.

 

Viewed from center of triangular prism

Related honeycombsEdit

Three runcinated regular compact honeycombs in H3
Image      
Symbols t0,3{4,3,5}
       
t0,3{3,5,3}
       
t0,3{5,3,5}
       
Vertex
figure
     

Runcitruncated icosahedral honeycombEdit

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  , [3,5,3]
Properties Vertex-transitive

The runcitruncated icosahedral honeycomb, t0,1,3{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

Related honeycombsEdit

Omnitruncated icosahedral honeycombEdit

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  , [[3,5,3]]
Properties Vertex-transitive

The omnitruncated icosahedral honeycomb, t0,1,2,3{3,5,3},        , has truncated icosidodecahedron and hexagonal prism cells, with a phyllic disphenoid vertex figure.

 

Centered on hexagonal prism

Related honeycombsEdit

Omnisnub icosahedral honeycombEdit

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(t0,1,2,3{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 honeycombEdit

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]

 

 

See alsoEdit

ReferencesEdit

  1. ^ Wendy Y. Krieger, Walls and bridges: The view from six dimensions, Symmetry: Culture and Science Volume 16, Number 2, pages 171–192 (2005) [1] Archived 2013-10-07 at the Wayback Machine
  2. ^ http://www.bendwavy.org/klitzing/incmats/pt353.htm
  • Coxeter, Regular Polytopes, 3rd. ed., Dover Publications, 1973. ISBN 0-486-61480-8. (Tables I and II: Regular polytopes and honeycombs, pp. 294–296)
  • Coxeter, The Beauty of Geometry: Twelve Essays, Dover Publications, 1999 ISBN 0-486-40919-8 (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
  • Klitzing, Richard. "Hyperbolic H3 honeycombs hyperbolic order 3 icosahedral tesselation".