Hexagonal tiling honeycomb

In the field of hyperbolic geometry, the hexagonal tiling honeycomb is one of 11 regular paracompact honeycombs in 3-dimensional hyperbolic space. It is paracompact because it has cells composed of an infinite number of faces. Each cell is a hexagonal tiling whose vertices lie on a horosphere, a surface in hyperbolic space that approaches a single ideal point at infinity.

Hexagonal tiling honeycomb

Perspective projection view
within Poincaré disk model
TypeHyperbolic regular honeycomb
Paracompact uniform honeycomb
Schläfli symbols{6,3,3}
t{3,6,3}
2t{6,3,6}
2t{6,3[3]}
t{3[3,3]}
Coxeter diagrams




Cells{6,3}
Faceshexagon {6}
Edge figuretriangle {3}
Vertex figure
tetrahedron {3,3}
DualOrder-6 tetrahedral honeycomb
Coxeter groups, [3,3,6]
, [3,6,3]
, [6,3,6]
, [6,3[3]]
, [3[3,3]]
PropertiesRegular

The Schläfli symbol of the hexagonal tiling honeycomb is {6,3,3}. Since that of the hexagonal tiling is {6,3}, this honeycomb has three such hexagonal tilings meeting at each edge. Since the Schläfli symbol of the tetrahedron is {3,3}, the vertex figure of this honeycomb is a tetrahedron. Thus, four hexagonal tilings meet at each vertex of this honeycomb, six hexagons meet at each vertex, and four edges meet at each vertex.[1]

Images

Viewed in perspective outside of a Poincaré disk model, the image above shows one hexagonal tiling cell within the honeycomb, and its mid-radius horosphere (the horosphere incident with edge midpoints). In this projection, the hexagons grow infinitely small towards the infinite boundary, asymptoting towards a single ideal point. It can be seen as similar to the order-3 apeirogonal tiling, {,3} of H2, with horocycles circumscribing vertices of apeirogonal faces.

{6,3,3} {,3}
One hexagonal tiling cell of the hexagonal tiling honeycomb An order-3 apeirogonal tiling with a green apeirogon and its horocycle

Symmetry constructions

It has a total of five reflectional constructions from five related Coxeter groups all with four mirrors and only the first being regular: [6,3,3], [3,6,3], [6,3,6], [6,3[3]] and [3[3,3]] , having 1, 4, 6, 12 and 24 times larger fundamental domains respectively. In Coxeter notation subgroup markups, they are related as: [6,(3,3)*] (remove 3 mirrors, index 24 subgroup); [3,6,3*] or [3*,6,3] (remove 2 mirrors, index 6 subgroup); [1+,6,3,6,1+] (remove two orthogonal mirrors, index 4 subgroup); all of these are isomorphic to [3[3,3]]. The ringed Coxeter diagrams are , , , and , representing different types (colors) of hexagonal tilings in the Wythoff construction.

The hexagonal tiling honeycomb is a regular hyperbolic honeycomb in 3-space, and one of 11 which are paracompact.

11 paracompact regular honeycombs

{6,3,3}

{6,3,4}

{6,3,5}

{6,3,6}

{4,4,3}

{4,4,4}

{3,3,6}

{4,3,6}

{5,3,6}

{3,6,3}

{3,4,4}

It is one of 15 uniform paracompact honeycombs in the [6,3,3] Coxeter group, along with its dual, the order-6 tetrahedral honeycomb.

It is part of a sequence of regular polychora, which include the 5-cell {3,3,3}, tesseract {4,3,3}, and 120-cell {5,3,3} of Euclidean 4-space, along with other hyperbolic honeycombs containing tetrahedral vertex figures.

It is also part of a sequence of regular honeycombs of the form {6,3,p}, which are each composed of hexagonal tiling cells:

Rectified hexagonal tiling honeycomb

Rectified hexagonal tiling honeycomb
TypeParacompact uniform honeycomb
Schläfli symbolsr{6,3,3} or t1{6,3,3}
Coxeter diagrams
Cells{3,3}
r{6,3} or
Facestriangle {3}
hexagon {6}
Vertex figure
triangular prism
Coxeter groups, [3,3,6]
, [3,3[3]]
PropertiesVertex-transitive, edge-transitive

The rectified hexagonal tiling honeycomb, t1{6,3,3}, has tetrahedral and trihexagonal tiling facets, with a triangular prism vertex figure. The half-symmetry construction alternates two types of tetrahedra.

Hexagonal tiling honeycomb
Rectified hexagonal tiling honeycomb
or
Related H2 tilings
Order-3 apeirogonal tiling
Triapeirogonal tiling
or

Truncated hexagonal tiling honeycomb

Truncated hexagonal tiling honeycomb
TypeParacompact uniform honeycomb
Schläfli symbolt{6,3,3} or t0,1{6,3,3}
Coxeter diagram
Cells{3,3}
t{6,3}
Facestriangle {3}
dodecagon {12}
Vertex figure
triangular pyramid
Coxeter groups, [3,3,6]
PropertiesVertex-transitive

The truncated hexagonal tiling honeycomb, t0,1{6,3,3}, has tetrahedral and truncated hexagonal tiling facets, with a triangular pyramid vertex figure.

It is similar to the 2D hyperbolic truncated order-3 apeirogonal tiling, t{,3} with apeirogonal and triangle faces:

Bitruncated hexagonal tiling honeycomb

Bitruncated hexagonal tiling honeycomb
Bitruncated order-6 tetrahedral honeycomb
TypeParacompact uniform honeycomb
Schläfli symbol2t{6,3,3} or t1,2{6,3,3}
Coxeter diagram
Cellst{3,3}
t{3,6}
Facestriangle {3}
hexagon {6}
Vertex figure
digonal disphenoid
Coxeter groups, [3,3,6]
, [3,3[3]]
PropertiesVertex-transitive

The bitruncated hexagonal tiling honeycomb or bitruncated order-6 tetrahedral honeycomb, t1,2{6,3,3}, has truncated tetrahedron and hexagonal tiling cells, with a digonal disphenoid vertex figure.

Cantellated hexagonal tiling honeycomb

Cantellated hexagonal tiling honeycomb
TypeParacompact uniform honeycomb
Schläfli symbolrr{6,3,3} or t0,2{6,3,3}
Coxeter diagram
Cellsr{3,3}
rr{6,3}
{}×{3}
Facestriangle {3}
square {4}
hexagon {6}
Vertex figure
wedge
Coxeter groups, [3,3,6]
PropertiesVertex-transitive

The cantellated hexagonal tiling honeycomb, t0,2{6,3,3}, has octahedron, rhombitrihexagonal tiling, and triangular prism cells, with a wedge vertex figure.

Cantitruncated hexagonal tiling honeycomb

Cantitruncated hexagonal tiling honeycomb
TypeParacompact uniform honeycomb
Schläfli symboltr{6,3,3} or t0,1,2{6,3,3}
Coxeter diagram
Cellst{3,3}
tr{6,3}
{}×{3}
Facestriangle {3}
square {4}
hexagon {6}
dodecagon {12}
Vertex figure
mirrored sphenoid
Coxeter groups, [3,3,6]
PropertiesVertex-transitive

The cantitruncated hexagonal tiling honeycomb, t0,1,2{6,3,3}, has truncated tetrahedron, truncated trihexagonal tiling, and triangular prism cells, with a mirrored sphenoid vertex figure.

Runcinated hexagonal tiling honeycomb

Runcinated hexagonal tiling honeycomb
TypeParacompact uniform honeycomb
Schläfli symbolt0,3{6,3,3}
Coxeter diagram
Cells{3,3}
{6,3}
{}×{6}
{}×{3}
Facestriangle {3}
square {4}
hexagon {6}
Vertex figure
irregular triangular antiprism
Coxeter groups, [3,3,6]
PropertiesVertex-transitive

The runcinated hexagonal tiling honeycomb, t0,3{6,3,3}, has tetrahedron, hexagonal tiling, hexagonal prism, and triangular prism cells, with an irregular triangular antiprism vertex figure.

Runcitruncated hexagonal tiling honeycomb

Runcitruncated hexagonal tiling honeycomb
TypeParacompact uniform honeycomb
Schläfli symbolt0,1,3{6,3,3}
Coxeter diagram
Cellsrr{3,3}
{}x{3}
{}x{12}
t{6,3}
Facestriangle {3}
square {4}
hexagon {6}
dodecagon {12}
Vertex figure
isosceles-trapezoidal pyramid
Coxeter groups, [3,3,6]
PropertiesVertex-transitive

The runcitruncated hexagonal tiling honeycomb, t0,1,3{6,3,3}, has cuboctahedron, triangular prism, dodecagonal prism, and truncated hexagonal tiling cells, with an isosceles-trapezoidal pyramid vertex figure.

Runcicantellated hexagonal tiling honeycomb

Runcicantellated hexagonal tiling honeycomb
runcitruncated order-6 tetrahedral honeycomb
TypeParacompact uniform honeycomb
Schläfli symbolt0,2,3{6,3,3}
Coxeter diagram
Cellst{3,3}
{}x{6}
rr{6,3}
Facestriangle {3}
square {4}
hexagon {6}
Vertex figure
isosceles-trapezoidal pyramid
Coxeter groups, [3,3,6]
PropertiesVertex-transitive

The runcicantellated hexagonal tiling honeycomb or runcitruncated order-6 tetrahedral honeycomb, t0,2,3{6,3,3}, has truncated tetrahedron, hexagonal prism, and rhombitrihexagonal tiling cells, with an isosceles-trapezoidal pyramid vertex figure.

Omnitruncated hexagonal tiling honeycomb

Omnitruncated hexagonal tiling honeycomb
Omnitruncated order-6 tetrahedral honeycomb
TypeParacompact uniform honeycomb
Schläfli symbolt0,1,2,3{6,3,3}
Coxeter diagram
Cellstr{3,3}
{}x{6}
{}x{12}
tr{6,3}
Facessquare {4}
hexagon {6}
dodecagon {12}
Vertex figure
irregular tetrahedron
Coxeter groups, [3,3,6]
PropertiesVertex-transitive

The omnitruncated hexagonal tiling honeycomb or omnitruncated order-6 tetrahedral honeycomb, t0,1,2,3{6,3,3}, has truncated octahedron, hexagonal prism, dodecagonal prism, and truncated trihexagonal tiling cells, with an irregular tetrahedron vertex figure.

See also

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 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)
  • N. W. Johnson, R. Kellerhals, J. G. Ratcliffe, S. T. Tschantz, The size of a hyperbolic Coxeter simplex, Transformation Groups (1999), Volume 4, Issue 4, pp 329–353
  • N. W. Johnson, R. Kellerhals, J. G. Ratcliffe, S. T. Tschantz, Commensurability classes of hyperbolic Coxeter groups, (2002) H3: p130.
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