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词条 Order-5 hexagonal tiling honeycomb
释义

  1. Symmetry

  2. Images

  3. Related polytopes and honeycombs

      Rectified order-5 hexagonal tiling honeycomb    Truncated order-5 hexagonal tiling honeycomb    Cantellated order-5 hexagonal tiling honeycomb    Bitruncated order-5 hexagonal tiling honeycomb    Cantitruncated order-5 hexagonal tiling honeycomb    Runcinated order-5 hexagonal tiling honeycomb    Runcitruncated order-5 hexagonal tiling honeycomb    Omnitruncated order-5 hexagonal tiling honeycomb  

  4. See also

  5. References

Order-5 hexagonal tiling honeycomb

Perspective projection view
from center of Poincaré disk model
TypeHyperbolic regular honeycomb
Paracompact uniform honeycomb
Schläfli symbol{6,3,5}
Coxeter-Dynkin diagramsnode_1|6|node|3|node|5|node}}
↔ {{CDD|node_1|6|node_g|3sg|node_g|5g|node_g}}
Cells{6,3}
Faceshexagon {6}
Edge figurepentagon {5}
Vertex figure{3,5}
DualOrder-6 dodecahedral honeycomb
Coxeter groupHV}}3, [6,3,5]
PropertiesRegular

In the field of hyperbolic geometry, the order-5 hexagonal tiling honeycomb arises one of 11 regular paracompact honeycombs in 3-dimensional hyperbolic space. It is called paracompact because it has infinite cells. Each cell consists of a hexagonal tiling whose vertices lie on a horosphere: a flat plane in hyperbolic space that approaches a single ideal point at infinity.

The Schläfli symbol of the order-5 hexagonal tiling honeycomb is {6,3,5}. Since that of the hexagonal tiling of the plane is {6,3}, this honeycomb has five such hexagonal tilings meeting at each edge. Since the Schläfli symbol of the icosahedron is {3,5}, the vertex figure of this honeycomb is an icosahedron. Thus, 20 hexagonal tilings meet at each vertex of this honeycomb.[1]

{{Honeycomb}}

Symmetry

A lower symmetry, [6,(3,5)*], index 120 construction exists with regular dodecahedral fundamental domains, and an icosahedral shaped Coxeter diagram with 6 axial infinite order (ultraparallel) branches.

Images

It is similar to the 2D hyperbolic regular tiling, {∞,5}, with infinite apeirogonal faces, and 5 meeting around every vertex (peak).

Related polytopes and honeycombs

It is one of 15 regular hyperbolic honeycombs in 3-space, 11 of which like this one are paracompact, with infinite cells or vertex figures.

{{Regular_paracompact_H3_honeycombs}}

There are 15 uniform honeycombs in the [5,3,6] Coxeter group family, including this regular form and its regular dual, order-5 hexagonal tiling honeycomb, {6,3,5}.

{{635 family}}

It has a related alternation honeycomb, represented by {{CDD|node_h1|6|node|3|node|5|node}} ↔ {{CDD|branch_10ru|split2|node|5|node}}, having icosahedron and triangular tiling cells.

It is a part of sequence of regular honeycombs with hexagonal tiling hyperbolic honeycombs of the form {6,3,p}:

{{Hexagonal tiling cell tessellations}}

This honeycomb is a part of a sequence of polychora and honeycombs with icosahedron vertex figures:

{{Icosahedral vertex figure tessellations}}

Rectified order-5 hexagonal tiling honeycomb

Rectified order-5 hexagonal tiling honeycomb
TypeParacompact uniform honeycomb
Schläfli symbolsr{6,3,5} or t1{6,3,5}
2r{5,3[3]}
Coxeter diagramsnode|6|node_1|3|node|5|node}}
{{CDD|node_h0|6|node_1|3|node|5|node}} ↔ {{CDD|branch_11|split2|node|5|node}}
Cells{3,5}
r{6,3}, r{3[3]}
FacesTriangle {3}
Pentagon {5}
Hexagon {6}
Vertex figure
Pentagonal prism {}×{5}
Coxeter groups, [6,3,5]
PropertiesVertex-transitive, edge-transitive

The rectified order-5 hexagonal tiling honeycomb, t1{6,3,5}, {{CDD|node|6|node_1|3|node|5|node}} has icosahedron and trihexagonal tiling facets, with a pentagonal prism vertex figure.

It is similar to the 2D hyperbolic infinite-order square tiling, r{∞,5} with pentagon and apeirogonal faces. All vertices are on the ideal surface.

{{Pentagonal prism vertex figure tessellations}}{{-}}

Truncated order-5 hexagonal tiling honeycomb

Truncated order-5 hexagonal tiling honeycomb
TypeParacompact uniform honeycomb
Schläfli symbolt{6,3,5} or t0,1{6,3,5}
Coxeter diagramnode_1|6|node_1|3|node|5|node}}
Cells{3,5}
t{6,3}
FacesTriangle {3}
Pentagon {5}
Hexagon {6}
Vertex figure
Pentagonal pyramid {}v{5}
Coxeter groups, [6,3,5]
PropertiesVertex-transitive

The truncated order-5 hexagonal tiling honeycomb, t0,1{6,3,5}, {{CDD|node_1|6|node_1|3|node|5|node}} has icosahedron and triangular tiling facets, with a pentagonal pyramid vertex figure.

{{-}}

Cantellated order-5 hexagonal tiling honeycomb

Cantellated order-5 hexagonal tiling honeycomb
TypeParacompact uniform honeycomb
Schläfli symbolrr{6,3,5} or t0,2{6,3,5}
Coxeter diagramnode_1|6|node|3|node_1|5|node}}
Cellsr{3,5}
rr{6,3}
FacesTriangle {3}
Pentagon {5}
Hexagon {6}
Vertex figure
triangular prism
Coxeter groups, [6,3,5]
PropertiesVertex-transitive

The cantellated order-5 hexagonal tiling honeycomb, t0,2{6,3,5}, {{CDD|node_1|6|node|3|node_1|5|node}} has icosidodecahedron and rhombitrihexagonal tiling facets, with a triangular prism vertex figure.

{{-}}

Bitruncated order-5 hexagonal tiling honeycomb

Bitruncated order-5 hexagonal tiling honeycomb
TypeParacompact uniform honeycomb
Schläfli symbol2t{6,3,5} or t1,2{6,3,5}
Coxeter diagramnode|6|node_1|3|node_1|5|node}}
Cells
Faces
Vertex figure
Tetrahedron
Coxeter groups, [6,3,5]
PropertiesVertex-transitive

The bitruncated order-5 hexagonal tiling honeycomb, t1,2{6,3,5}, {{CDD|node|6|node_1|3|node_1|5|node}} has a tetrahedral vertex figure.

{{-}}

Cantitruncated order-5 hexagonal tiling honeycomb

Cantitruncated order-5 hexagonal tiling honeycomb
TypeParacompact uniform honeycomb
Schläfli symboltr{6,3,5} or t0,1,2{6,3,5}
Coxeter diagramnode_1|6|node_1|3|node_1|5|node}}
Cellst{3,5}
tr{6,3}
FacesPentagon {5}
Hexagon {6}
Dodecagon {12}
Vertex figure
triangular prism
Coxeter groups, [6,3,5]
PropertiesVertex-transitive

The cantitruncated order-5 hexagonal tiling honeycomb, t0,1,2{6,3,5}, {{CDD|node_1|6|node_1|3|node_1|5|node}} has truncated icosahedron and truncated trihexagonal tiling facets, with a tetrahedral vertex figure.

{{-}}

Runcinated order-5 hexagonal tiling honeycomb

Runcinated order-5 hexagonal tiling honeycomb
TypeParacompact uniform honeycomb
Schläfli symbolt0,3{6,3,5}
Coxeter diagramnode_1|6|node|3|node|5|node_1}}
Cells
Faces
Vertex figure
triangular antiprism
Coxeter groups, [6,3,5]
PropertiesVertex-transitive

The runcinated order-5 hexagonal tiling honeycomb, t0,3{6,3,5}, {{CDD|node_1|6|node|3|node|5|node_1}} has dodecahedron and truncated trihexagonal tiling facets, with a triangular antiprism vertex figure.

{{-}}

Runcitruncated order-5 hexagonal tiling honeycomb

Runcitruncated order-5 hexagonal tiling honeycomb
TypeParacompact uniform honeycomb
Schläfli symbolt0,1,3{6,3,5}
Coxeter diagramnode_1|6|node_1|3|node|5|node_1}}
Cells
Faces
Vertex figure
trapezoidal pyramid
Coxeter groups, [6,3,5]
PropertiesVertex-transitive

The runcitruncated order-5 hexagonal tiling honeycomb, t0,1,3{6,3,5}, {{CDD|node_1|6|node_1|3|node|5|node_1}} has a trapezoidal pyramid vertex figure.

{{-}}

Omnitruncated order-5 hexagonal tiling honeycomb

Omnitruncated order-5 hexagonal tiling honeycomb
TypeParacompact uniform honeycomb
Schläfli symbolt0,1,2,3{6,3,5}
Coxeter diagramnode_1|6|node_1|3|node_1|5|node_1}}
Cells
Faces
Vertex figure
tetrahedron
Coxeter groups, [6,3,5]
PropertiesVertex-transitive

The omnitruncated order-5 hexagonal tiling honeycomb, t0,1,2,3{6,3,5}, {{CDD|node_1|6|node_1|3|node_1|5|node_1}} has a tetrahedral vertex figure.

{{-}}

See also

  • Convex uniform honeycombs in hyperbolic space
  • List of regular polytopes

References

1. ^Coxeter The Beauty of Geometry, 1999, Chapter 10, Table III
  • 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|99035678}}, {{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}} (Chapter 16-17: Geometries on Three-manifolds I,II)
  • 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

2 : Hexagonal tilings|Honeycombs (geometry)
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