Order-4 icosahedral honeycomb

In the geometry of hyperbolic 3-space, the order-4 icosahedral honeycomb is a regular space-filling tessellation (or honeycomb) with Schläfli symbol {3,5,4}.

Order-4 icosahedral honeycomb
TypeRegular honeycomb
Schläfli symbols{3,5,4}
Coxeter diagrams
Cells{3,5}
Faces{3}
Edge figure{4}
Vertex figure{5,4}
Dual{4,5,3}
Coxeter group[3,5,4]
PropertiesRegular

Geometry

It has four icosahedra {3,5} around each edge. All vertices are ultra-ideal (existing beyond the ideal boundary) with infinitely many icosahedra existing around each vertex in an order-4 pentagonal tiling vertex arrangement.


Poincaré disk model
(Cell centered)

Ideal surface

It has a second construction as a uniform honeycomb, Schläfli symbol {3,51,1}, Coxeter diagram, , with alternating types or colors of icosahedral cells. In Coxeter notation the half symmetry is [3,5,4,1+] = [3,51,1].

It a part of a sequence of regular polychora and honeycombs with icosahedral cells: {3,5,p}

{3,5,p} polytopes
Space H3
Form Compact Noncompact
Name {3,5,3}

 
{3,5,4}
{3,5,5}
{3,5,6}
{3,5,7}
{3,5,8}
... {3,5,}
Image
Vertex
figure

{5,3}

{5,4}

{5,5}

{5,6}

{5,7}

{5,8}

{5,}

Order-5 icosahedral honeycomb

Order-5 icosahedral honeycomb
TypeRegular honeycomb
Schläfli symbols{3,5,5}
Coxeter diagrams
Cells{3,5}
Faces{3}
Edge figure{5}
Vertex figure{5,5}
Dual{5,5,3}
Coxeter group[3,5,5]
PropertiesRegular

In the geometry of hyperbolic 3-space, the order-5 icosahedral honeycomb is a regular space-filling tessellation (or honeycomb) with Schläfli symbol {3,5,5}. It has five icosahedra, {3,5}, around each edge. All vertices are ultra-ideal (existing beyond the ideal boundary) with infinitely many icosahedra existing around each vertex in an order-5 pentagonal tiling vertex arrangement.


Poincaré disk model
(Cell centered)

Ideal surface

Order-6 icosahedral honeycomb

Order-6 icosahedral honeycomb
TypeRegular honeycomb
Schläfli symbols{3,5,6}
{3,(5,∞,5)}
Coxeter diagrams
=
Cells{3,5}
Faces{3}
Edge figure{6}
Vertex figure{5,6}
Dual{6,5,3}
Coxeter group[3,5,6]
PropertiesRegular

In the geometry of hyperbolic 3-space, the order-6 icosahedral honeycomb is a regular space-filling tessellation (or honeycomb) with Schläfli symbol {3,5,6}. It has six icosahedra, {3,5}, around each edge. All vertices are ultra-ideal (existing beyond the ideal boundary) with infinitely many icosahedra existing around each vertex in an order-6 pentagonal tiling vertex arrangement.


Poincaré disk model
(Cell centered)

Ideal surface

Order-7 icosahedral honeycomb

Order-7 icosahedral honeycomb
TypeRegular honeycomb
Schläfli symbols{3,5,7}
Coxeter diagrams
Cells{3,5}
Faces{3}
Edge figure{7}
Vertex figure{5,7}
Dual{7,5,3}
Coxeter group[3,5,7]
PropertiesRegular

In the geometry of hyperbolic 3-space, the order-7 icosahedral honeycomb is a regular space-filling tessellation (or honeycomb) with Schläfli symbol {3,5,7}. It has seven icosahedra, {3,5}, around each edge. All vertices are ultra-ideal (existing beyond the ideal boundary) with infinitely many icosahedra existing around each vertex in an order-7 pentagonal tiling vertex arrangement.


Poincaré disk model
(Cell centered)

Ideal surface

Order-8 icosahedral honeycomb

Order-8 icosahedral honeycomb
TypeRegular honeycomb
Schläfli symbols{3,5,8}
Coxeter diagrams
Cells{3,5}
Faces{3}
Edge figure{8}
Vertex figure{5,8}
Dual{8,5,3}
Coxeter group[3,5,8]
PropertiesRegular

In the geometry of hyperbolic 3-space, the order-8 icosahedral honeycomb is a regular space-filling tessellation (or honeycomb) with Schläfli symbol {3,5,8}. It has eight icosahedra, {3,5}, around each edge. All vertices are ultra-ideal (existing beyond the ideal boundary) with infinitely many icosahedra existing around each vertex in an order-8 pentagonal tiling vertex arrangement.


Poincaré disk model
(Cell centered)

Infinite-order icosahedral honeycomb

Infinite-order icosahedral honeycomb
TypeRegular honeycomb
Schläfli symbols{3,5,∞}
{3,(5,∞,5)}
Coxeter diagrams
=
Cells{3,5}
Faces{3}
Edge figure{∞}
Vertex figure{5,∞}
{(5,∞,5)}
Dual{∞,5,3}
Coxeter group[∞,5,3]
[3,((5,∞,5))]
PropertiesRegular

In the geometry of hyperbolic 3-space, the infinite-order icosahedral honeycomb is a regular space-filling tessellation (or honeycomb) with Schläfli symbol {3,5,∞}. It has infinitely many icosahedra, {3,5}, around each edge. All vertices are ultra-ideal (existing beyond the ideal boundary) with infinitely many icosahedra existing around each vertex in an infinite-order triangular tiling vertex arrangement.


Poincaré disk model
(Cell centered)

Ideal surface

It has a second construction as a uniform honeycomb, Schläfli symbol {3,(5,∞,5)}, Coxeter diagram, = , with alternating types or colors of icosahedral cells. In Coxeter notation the half symmetry is [3,5,∞,1+] = [3,((5,∞,5))].

See also

References

    • Coxeter, Regular Polytopes, 3rd. ed., Dover Publications, 1973. ISBN 0-486-61480-8. (Tables I and II: Regular polytopes and honeycombs, pp. 294–296)
    • The Beauty of Geometry: Twelve Essays (1999), Dover Publications, LCCN 99-35678, ISBN 0-486-40919-8 (Chapter 10, Regular Honeycombs in Hyperbolic Space) Table III
    • Jeffrey R. Weeks The Shape of Space, 2nd edition ISBN 0-8247-0709-5 (Chapters 16–17: Geometries on Three-manifolds I,II)
    • George Maxwell, Sphere Packings and Hyperbolic Reflection Groups, JOURNAL OF ALGEBRA 79,78-97 (1982)
    • Hao Chen, Jean-Philippe Labbé, Lorentzian Coxeter groups and Boyd-Maxwell ball packings, (2013)
    • Visualizing Hyperbolic Honeycombs arXiv:1511.02851 Roice Nelson, Henry Segerman (2015)
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