Hexagonal tiling honeycomb

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Hexagonal tiling honeycomb
H3 633 FC boundary.png
Perspective projection view
within Poincaré disk model
Type Hyperbolic 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 CDel node 1.pngCDel 6.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
CDel node 1.pngCDel 3.pngCDel node 1.pngCDel 6.pngCDel node.pngCDel 3.pngCDel node.png
CDel node.pngCDel 6.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 6.pngCDel node.png
CDel branch 11.pngCDel split2.pngCDel node 1.pngCDel 6.pngCDel node.png
CDel branch 11.pngCDel splitcross.pngCDel branch 11.pngCDel node 1.pngCDel 6.pngCDel node g.pngCDel 3sg.pngCDel node g.pngCDel 3g.pngCDel node g.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 6.pngCDel node g.pngCDel 3sg.pngCDel node g.png
CDel node h0.pngCDel 6.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 6.pngCDel node h0.pngCDel branch 11.pngCDel split2.pngCDel node 1.pngCDel 6.pngCDel node h0.png
Cells {6,3} Uniform tiling 63-t0.png
Faces Hexagon {6}
Edge figure Triangle {3}
Vertex figure Order-3 hexagonal tiling honeycomb verf.png
tetrahedron {3,3}
Dual {3,3,6}
Coxeter groups , [6,3,3]
, [3,6,3]
, [6,3,6]
, [6,3[3]]
, [3[3,3]]
Properties Regular

In the field of hyperbolic geometry, the 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 hexagonal tiling honeycomb is {6,3,3}. Since that of the hexagonal tiling of the plane 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 an tetrahedron. Thus, six hexagonal tilings meet at each vertex of this honeycomb, and four edges meet at each vertex.[1]

Images[edit]

H3 363-1100.png

Viewed in perspective outside of a Poincaré disk model, this 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 horocycle circumscribing vertices of apeirogonal faces.

{6,3,3} {∞,3}
633 honeycomb one cell horosphere.png Order-3 apeirogonal tiling one cell horocycle.png
One hexagonal tiling of this honeycomb order-3 apeirogonal tiling with a green apeirogon and its horocycle

Symmetry constructions[edit]

It has a total of five reflectional constructions from five related Coxeter groups all with four mirrors and only the first being regular: CDel node c1.pngCDel 6.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png [6,3,3], CDel node c1.pngCDel 3.pngCDel node c1.pngCDel 6.pngCDel node.pngCDel 3.pngCDel node.png [3,6,3], CDel node.pngCDel 6.pngCDel node c1.pngCDel 3.pngCDel node c1.pngCDel 6.pngCDel node.png [6,3,6], CDel branch c1.pngCDel split2.pngCDel node c1.pngCDel 6.pngCDel node.png [6,3[3]] and [3[3,3]] CDel branch c1.pngCDel splitcross.pngCDel branch c1.png, 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 CDel node 1.pngCDel 6.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png, CDel node 1.pngCDel 3.pngCDel node 1.pngCDel 6.pngCDel node.pngCDel 3.pngCDel node.png, CDel node.pngCDel 6.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 6.pngCDel node.png, CDel branch 11.pngCDel split2.pngCDel node 1.pngCDel 6.pngCDel node.png and CDel branch 11.pngCDel splitcross.pngCDel branch 11.png, representing different types (colors) of hexagonal tilings in the Wythoff construction.

Related polytopes and honeycombs[edit]

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.

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

Polytopes and honeycombs with tetrahedral vertex figures[edit]

It is in a sequence with regular polychora: 5-cell {3,3,3}, tesseract {4,3,3}, 120-cell {5,3,3} of Euclidean 4-space, with tetrahedral vertex figures.


Polytopes and honeycombs with hexagonal tiling cells[edit]

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

Rectified hexagonal tiling honeycomb[edit]

Rectified hexagonal tiling honeycomb
H3 633 boundary 0100.png
Type Paracompact uniform honeycomb
Schläfli symbols r{6,3,3} or t1{6,3,3}
Coxeter diagrams CDel node.pngCDel 6.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
CDel node h0.pngCDel 6.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel branch 11.pngCDel split2.pngCDel node.pngCDel 3.pngCDel node.png
Cells {3,3} Uniform polyhedron-33-t2.png
r{6,3} Uniform tiling 63-t1.png or Uniform tiling 333-t12.png
Faces Triangle {3}
Hexagon {6}
Vertex figure Rectified order-3 hexagonal tiling honeycomb verf.png
Triangular prism {}×{3}
Coxeter groups , [6,3,3]
Properties Vertex-transitive, edge-transitive

The rectified hexagonal tiling honeycomb, t1{6,3,3}, CDel node.pngCDel 6.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png has tetrahedral and trihexagonal tiling facets, with a triangular prism vertex figure. The CDel branch 11.pngCDel split2.pngCDel node.pngCDel 3.pngCDel node.png half-symmetry construction alternate two types of tetrahedra.

Hexagonal tiling honeycomb
CDel node 1.pngCDel 6.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
Rectified hexagonal tiling honeycomb
CDel node.pngCDel 6.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png or CDel branch 11.pngCDel split2.pngCDel node.pngCDel 3.pngCDel node.png
Hyperbolic 3d hexagonal tiling.png Hyperbolic 3d rectified hexagonal tiling.png
Related H2 tilings
Order-3 apeirogonal tiling
CDel node 1.pngCDel infin.pngCDel node.pngCDel 3.pngCDel node.png
Triapeirogonal tiling
CDel node.pngCDel infin.pngCDel node 1.pngCDel 3.pngCDel node.png or CDel labelinfin.pngCDel branch 11.pngCDel split2.pngCDel node.png
H2 tiling 23i-1.png H2 tiling 23i-2.pngH2 tiling 33i-3.png

Truncated hexagonal tiling honeycomb[edit]

Truncated hexagonal tiling honeycomb
Type Paracompact uniform honeycomb
Schläfli symbol t{6,3,3} or t0,1{6,3,3}
Coxeter diagram CDel node 1.pngCDel 6.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
Cells {3,3} Uniform polyhedron-33-t2.png
t{6,3} Uniform tiling 63-t01.png
Faces Triangle {3}
Dodecagon {12}
Vertex figure Truncated order-3 hexagonal tiling honeycomb verf.png
tetrahedron
Coxeter groups , [6,3,3]
Properties Vertex-transitive

The truncated hexagonal tiling honeycomb, t0,1{6,3,3}, CDel node 1.pngCDel 6.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png has tetrahedral and truncated hexagonal tiling facets, with a tetrahedral vertex figure.

H3 633-1100.png

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

H2 tiling 23i-3.png

Bitruncated hexagonal tiling honeycomb[edit]

Bitruncated hexagonal tiling honeycomb
Bitruncated order-6 tetrahedral honeycomb
Type Paracompact uniform honeycomb
Schläfli symbol 2t{6,3,3} or t1,2{6,3,3}
Coxeter diagram CDel node.pngCDel 6.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.png
CDel branch 11.pngCDel split2.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel node h0.pngCDel 6.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.png
Cells t{3,3} Uniform polyhedron-33-t01.png
t{3,6} Uniform tiling 63-t12.png
Faces Triangle {3}
hexagon {6}
Vertex figure Bitruncated order-3 hexagonal tiling honeycomb verf.png
tetrahedron
Coxeter groups , [6,3,3]
, [3,3[3]]
Properties Vertex-transitive

The bitruncated hexagonal tiling honeycomb or bitruncated order-6 tetrahedral honeycomb, t1,2{6,3,3}, CDel node.pngCDel 6.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.png has truncated tetrahedra and hexagonal tiling cells, with a tetrahedral vertex figure.

H3 633-0110.png

Cantellated hexagonal tiling honeycomb[edit]

Cantellated hexagonal tiling honeycomb
Type Paracompact uniform honeycomb
Schläfli symbol rr{6,3,3} or t0,2{6,3,3}
Coxeter diagram CDel node 1.pngCDel 6.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.png
Cells r{3,3} Uniform polyhedron-33-t1.png
rr{6,3} Uniform tiling 63-t02.png
{}×{3} Triangular prism.png
Faces Triangle {3}
Square {4}
Hexagon {6}
Vertex figure Cantellated order-3 hexagonal tiling honeycomb verf.png
Irreg. triangular prism
Coxeter groups , [6,3,3]
Properties Vertex-transitive

The cantellated hexagonal tiling honeycomb, t0,2{6,3,3}, CDel node 1.pngCDel 6.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.png has octahedral and rhombitrihexagonal tiling cells, with a triangular prism vertex figure.

H3 633-1010.png

Cantitruncated hexagonal tiling honeycomb[edit]

Cantitruncated hexagonal tiling honeycomb
Type Paracompact uniform honeycomb
Schläfli symbol tr{6,3,3} or t0,1,2{6,3,3}
Coxeter diagram CDel node 1.pngCDel 6.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.png
Cells t{3,3} Uniform polyhedron-33-t01.png
tr{6,3} Uniform tiling 63-t012.png
{}×{3} Triangular prism.png
Faces Triangle {3}
Square {4}
Hexagon {6}
Vertex figure Cantitruncated order-3 hexagonal tiling honeycomb verf.png
Irreg. tetrahedron
Coxeter groups , [6,3,3]
Properties Vertex-transitive

The cantitruncated hexagonal tiling honeycomb, t0,1,2{6,3,3}, CDel node 1.pngCDel 6.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.png has truncated tetrahedron and truncated trihexagonal tiling cells, with a tetrahedron vertex figure.

H3 633-1110.png

Runcinated hexagonal tiling honeycomb[edit]

Runcinated hexagonal tiling honeycomb
Type Paracompact uniform honeycomb
Schläfli symbol t0,3{6,3,3}
Coxeter diagram CDel node 1.pngCDel 6.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.png
Cells {3,3} Uniform polyhedron-33-t0.png
t0,2{6,3} Uniform tiling 63-t02.png
{}×{6}Hexagonal prism.png
{}×{3} Triangular prism.png
Faces Triangle {3}
Square {4}
Hexagon {6}
Vertex figure Runcinated order-3 hexagonal tiling honeycomb verf.png
Octahedron
Coxeter groups , [6,3,3]
Properties Vertex-transitive

The runcinated hexagonal tiling honeycomb, t0,3{6,3,3}, CDel node 1.pngCDel 6.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.png has tetrahedron, rhombitrihexagonal tiling hexagonal prism, triangular prism cells, with a octahedron vertex figure.

H3 633-1001.png

Runcitruncated hexagonal tiling honeycomb[edit]

Runcitruncated hexagonal tiling honeycomb
Type Paracompact uniform honeycomb
Schläfli symbol t0,1,3{6,3,3}
Coxeter diagram CDel node 1.pngCDel 6.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.png
Cells rr{3,3} Uniform polyhedron-33-t02.png
{}x{3} Triangular prism.png
{}x{12} Dodecagonal prism.png
t{6,3} Uniform tiling 63-t01.png
Faces Triangle {3}
Square {4}
Hexagon {6}
Dodecagon {12}
Vertex figure Runcitruncated order-3 hexagonal tiling honeycomb verf.png
quad-pyramid
Coxeter groups , [6,3,3]
Properties Vertex-transitive

The runcitruncated hexagonal tiling honeycomb, t0,1,3{6,3,3}, CDel node 1.pngCDel 6.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.png has cuboctahedron, Triangular prism, Dodecagonal prism, and truncated hexagonal tiling cells, with a quad-pyramid vertex figure.

H3 633-1101.png

Runcicantellated hexagonal tiling honeycomb[edit]

Runcicantellated hexagonal tiling honeycomb
runcitruncated order-6 tetrahedral honeycomb
Type Paracompact uniform honeycomb
Schläfli symbol t0,2,3{6,3,3}
Coxeter diagram CDel node 1.pngCDel 6.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.png
Cells t{3,3} Uniform polyhedron-33-t12.png
{}x{6} Hexagonal prism.png
rr{6,3} Uniform tiling 63-t02.png
Faces Triangle {3}
Square {4}
Hexagon {6}
Vertex figure Runcitruncated order-6 tetrahedral honeycomb verf.png
quad-pyramid
Coxeter groups , [6,3,3]
Properties Vertex-transitive

The runcicantellated hexagonal tiling honeycomb or runcitruncated order-6 tetrahedral honeycomb, t0,2,3{6,3,3}, CDel node 1.pngCDel 6.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.png has truncated tetrahedron, hexagonal prism, hexagonal prism, and rhombitrihexagonal tiling cells, with a quad-pyramid vertex figure.

H3 633-1011.png

Omnitruncated hexagonal tiling honeycomb[edit]

Omnitruncated hexagonal tiling honeycomb
Omnitruncated order-6 tetrahedral honeycomb
Type Paracompact uniform honeycomb
Schläfli symbol t0,1,2,3{6,3,3}
Coxeter diagram CDel node 1.pngCDel 6.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.png
Cells tr{3,3} Uniform polyhedron-33-t012.png
{}x{6} Hexagonal prism.png
{}x{12} Dodecagonal prism.png
tr{6,3} Uniform tiling 63-t012.png
Faces Square {4}
Hexagon {6}
Dodecagon {12}
Vertex figure Omnitruncated order-3 hexagonal tiling honeycomb verf.png
tetrahedron
Coxeter groups , [6,3,3]
Properties Vertex-transitive

The omnitruncated hexagonal tiling honeycomb or omnitruncated order-6 tetrahedral honeycomb, t0,1,2,3{6,3,3}, CDel node 1.pngCDel 6.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.png has truncated octahedron, hexagonal prism, dodecagonal prism, and truncated trihexagonal tiling cells, with a quad-pyramid vertex figure.

H3 633-1111.png

See also[edit]

References[edit]

  1. ^ Coxeter The Beauty of Geometry, 1999, Chapter 10, Table III

External links[edit]