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 Order-6 tetrahedral honeycomb
Coxeter groups , [3,3,6]
, [3,6,3]
, [6,3,6]
, [6,3[3]]
, [3[3,3]]
PropertiesRegular

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.

Contents

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

H3 363-1100.png

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}
633 honeycomb one cell horosphere.png Order-3 apeirogonal tiling one cell horocycle.png
One hexagonal tiling cell of the hexagonal tiling honeycombAn order-3 apeirogonal tiling with a green apeirogon and its horocycle

Symmetry constructions

Subgroup relations Hyperbolic subgroup tree 336-direct.png
Subgroup relations

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.

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

11 paracompact regular honeycombs
H3 633 FC boundary.png
{6,3,3}
H3 634 FC boundary.png
{6,3,4}
H3 635 FC boundary.png
{6,3,5}
H3 636 FC boundary.png
{6,3,6}
H3 443 FC boundary.png
{4,4,3}
H3 444 FC boundary.png
{4,4,4}
H3 336 CC center.png
{3,3,6}
H3 436 CC center.png
{4,3,6}
H3 536 CC center.png
{5,3,6}
H3 363 FC boundary.png
{3,6,3}
H3 344 CC center.png
{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.

[6,3,3] family honeycombs
{6,3,3} r{6,3,3} t{6,3,3} rr{6,3,3} t0,3{6,3,3} tr{6,3,3} t0,1,3{6,3,3} t0,1,2,3{6,3,3}
H3 633 FC boundary.png H3 633 boundary 0100.png H3 633-1100.png H3 633-1010.png H3 633-1001.png H3 633-1110.png H3 633-1101.png H3 633-1111.png
H3 336 CC center.png H3 336 CC center 0100.png H3 633-0011.png H3 633-0101.png H3 633-0110.png H3 633-0111.png H3 633-1011.png
{3,3,6} r{3,3,6} t{3,3,6} rr{3,3,6} 2t{3,3,6} tr{3,3,6} t0,1,3{3,3,6} t0,1,2,3{3,3,6}

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.

{p,3,3} honeycombs
Space S3 H3
FormFiniteParacompactNoncompact
Name {3,3,3} {4,3,3} {5,3,3} {6,3,3} {7,3,3} {8,3,3} ... {,3,3}
Image Stereographic polytope 5cell.png Stereographic polytope 8cell.png Stereographic polytope 120cell faces.png H3 633 FC boundary.png Hyperbolic honeycomb 7-3-3 poincare.png Hyperbolic honeycomb 8-3-3 poincare.png Hyperbolic honeycomb i-3-3 poincare.png
Coxeter diagrams
Coxeter subgroup tree 33p-direct.png
1CDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel node 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel node 1.pngCDel 5.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel node 1.pngCDel 6.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel node 1.pngCDel 7.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel node 1.pngCDel 8.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel node 1.pngCDel infin.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
4CDel node 1.pngCDel 2.pngCDel node 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 6.pngCDel node.pngCDel 3.pngCDel node.pngCDel node 1.pngCDel 4.pngCDel node 1.pngCDel 8.pngCDel node.pngCDel 3.pngCDel node.pngCDel node 1.pngCDel infin.pngCDel node 1.pngCDel infin.pngCDel node.pngCDel 3.pngCDel node.png
6CDel node.pngCDel 4.pngCDel node 1.pngCDel 2.pngCDel node 1.pngCDel 4.pngCDel node.pngCDel node.pngCDel 6.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 6.pngCDel node.pngCDel node.pngCDel 8.pngCDel node 1.pngCDel 4.pngCDel node 1.pngCDel 8.pngCDel node.pngCDel node.pngCDel infin.pngCDel node 1.pngCDel infin.pngCDel node 1.pngCDel infin.pngCDel node.png
12CDel nodes 11.pngCDel 2.pngCDel node 1.pngCDel 4.pngCDel node.pngCDel branch 11.pngCDel split2.pngCDel node 1.pngCDel 6.pngCDel node.pngCDel label4.pngCDel branch 11.pngCDel split2-44.pngCDel node 1.pngCDel 8.pngCDel node.pngCDel labelinfin.pngCDel branch 11.pngCDel split2-ii.pngCDel node 1.pngCDel infin.pngCDel node.png
24CDel nodes 11.pngCDel 2.pngCDel nodes 11.pngCDel branch 11.pngCDel splitcross.pngCDel branch 11.png Cdel tet4 1111.png Cdel tetinfin 1111.png
Cells
{p,3}
CDel node 1.pngCDel p.pngCDel node.pngCDel 3.pngCDel node.png
Tetrahedron.png
{3,3}
CDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
Hexahedron.png
{4,3}
CDel node 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.png
CDel node.pngCDel 4.pngCDel node 1.pngCDel 2.pngCDel node 1.png
CDel nodes 11.pngCDel 2.pngCDel node 1.png
Dodecahedron.png
{5,3}
CDel node 1.pngCDel 5.pngCDel node.pngCDel 3.pngCDel node.png
Uniform tiling 63-t0.svg
{6,3}
CDel node 1.pngCDel 6.pngCDel node.pngCDel 3.pngCDel node.png
CDel node.pngCDel 6.pngCDel node 1.pngCDel 3.pngCDel node 1.png
CDel branch 11.pngCDel split2.pngCDel node 1.png
Heptagonal tiling.svg
{7,3}
CDel node 1.pngCDel 7.pngCDel node.pngCDel 3.pngCDel node.png
H2-8-3-dual.svg
{8,3}
CDel node 1.pngCDel 8.pngCDel node.pngCDel 3.pngCDel node.png
CDel node.pngCDel 8.pngCDel node 1.pngCDel 4.pngCDel node 1.png
CDel label4.pngCDel branch 11.pngCDel split2-44.pngCDel node 1.png
H2-I-3-dual.svg
{,3}
CDel node 1.pngCDel infin.pngCDel node.pngCDel 3.pngCDel node.png
CDel node.pngCDel infin.pngCDel node 1.pngCDel infin.pngCDel node 1.png
CDel labelinfin.pngCDel branch 11.pngCDel split2-ii.pngCDel node 1.png

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

{6,3,p} honeycombs
Space H3
Form Paracompact Noncompact
Name {6,3,3} {6,3,4} {6,3,5} {6,3,6} {6,3,7} {6,3,8} ... {6,3,}
Coxeter
CDel node 1.pngCDel 6.pngCDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node h0.png
CDel node 1.pngCDel 6.pngCDel node g.pngCDel 3sg.pngCDel node g.pngCDel 4.pngCDel node.png
CDel node 1.pngCDel 6.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel node 1.pngCDel 6.pngCDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.png
CDel node 1.pngCDel 6.pngCDel node.pngCDel split1.pngCDel nodes.png
CDel node.pngCDel ultra.pngCDel node 1.pngCDel split1.pngCDel branch 11.pngCDel uaub.pngCDel nodes.png
CDel node 1.pngCDel 6.pngCDel node.pngCDel 3.pngCDel node.pngCDel 5.pngCDel node.pngCDel node 1.pngCDel 6.pngCDel node.pngCDel 3.pngCDel node.pngCDel 6.pngCDel node.png
CDel node 1.pngCDel 6.pngCDel node.pngCDel split1.pngCDel branch.png
CDel node 1.pngCDel splitplit1u.pngCDel branch4u 11.pngCDel uabc.pngCDel branch4u.pngCDel splitplit2u.pngCDel node.png
CDel node 1.pngCDel 6.pngCDel node.pngCDel 3.pngCDel node.pngCDel 7.pngCDel node.pngCDel node 1.pngCDel 6.pngCDel node.pngCDel 3.pngCDel node.pngCDel 8.pngCDel node.png
CDel node 1.pngCDel 6.pngCDel node.pngCDel split1.pngCDel branch.pngCDel label4.png
CDel node 1.pngCDel 6.pngCDel node.pngCDel 3.pngCDel node.pngCDel infin.pngCDel node.png
CDel node 1.pngCDel 6.pngCDel node.pngCDel split1.pngCDel branch.pngCDel labelinfin.png
CDD 6-3star-infin.png
Image H3 633 FC boundary.png H3 634 FC boundary.png H3 635 FC boundary.png H3 636 FC boundary.png Hyperbolic honeycomb 6-3-7 poincare.png Hyperbolic honeycomb 6-3-8 poincare.png Hyperbolic honeycomb 6-3-i poincare.png
Vertex
figure
{3,p}
CDel node 1.pngCDel 3.pngCDel node.pngCDel p.pngCDel node.png
Tetrahedron.png
{3,3}
CDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
Octahedron.png
{3,4}
CDel node 1.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.png
CDel node 1.pngCDel split1.pngCDel nodes.png
Icosahedron.png
{3,5}
CDel node 1.pngCDel 3.pngCDel node.pngCDel 5.pngCDel node.png
Uniform tiling 63-t2.svg
{3,6}
CDel node 1.pngCDel 3.pngCDel node.pngCDel 6.pngCDel node.png
CDel node 1.pngCDel split1.pngCDel branch.png
Order-7 triangular tiling.svg
{3,7}
CDel node 1.pngCDel 3.pngCDel node.pngCDel 7.pngCDel node.png
H2-8-3-primal.svg
{3,8}
CDel node 1.pngCDel 3.pngCDel node.pngCDel 8.pngCDel node.png
CDel node 1.pngCDel split1.pngCDel branch.pngCDel label4.png
H2 tiling 23i-4.png
{3,}
CDel node 1.pngCDel 3.pngCDel node.pngCDel infin.pngCDel node.png
CDel node 1.pngCDel split1.pngCDel branch.pngCDel labelinfin.png

Rectified hexagonal tiling honeycomb

Rectified hexagonal tiling honeycomb
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
Coxeter groups , [3,3,6]
, [3,3[3]]
PropertiesVertex-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 alternates two types of tetrahedra.

H3 633 boundary 0100.png

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-I-3-dual.svg H2 tiling 23i-2.png H2 tiling 33i-3.png

Truncated hexagonal tiling honeycomb

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
triangular pyramid
Coxeter groups , [3,3,6]
PropertiesVertex-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 triangular pyramid 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

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
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}, CDel node.pngCDel 6.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.png has truncated tetrahedron and hexagonal tiling cells, with a digonal disphenoid vertex figure.

H3 633-0110.png

Cantellated hexagonal tiling honeycomb

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
wedge
Coxeter groups , [3,3,6]
PropertiesVertex-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 octahedron, rhombitrihexagonal tiling, and triangular prism cells, with a wedge vertex figure.

H3 633-1010.png

Cantitruncated hexagonal tiling honeycomb

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.svg
{}×{3} Triangular prism.png
Faces triangle {3}
square {4}
hexagon {6}
dodecagon {12}
Vertex figure Cantitruncated order-3 hexagonal tiling honeycomb verf.png
mirrored sphenoid
Coxeter groups , [3,3,6]
PropertiesVertex-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, truncated trihexagonal tiling, and triangular prism cells, with a mirrored sphenoid vertex figure.

H3 633-1110.png

Runcinated hexagonal tiling honeycomb

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
{6,3} Uniform tiling 63-t0.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
irregular triangular antiprism
Coxeter groups , [3,3,6]
PropertiesVertex-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, hexagonal tiling, hexagonal prism, and triangular prism cells, with an irregular triangular antiprism vertex figure.

H3 633-1001.png

Runcitruncated hexagonal tiling honeycomb

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}
dodecagon {12}
Vertex figure Runcitruncated order-3 hexagonal tiling honeycomb verf.png
isosceles-trapezoidal pyramid
Coxeter groups , [3,3,6]
PropertiesVertex-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 an isosceles-trapezoidal pyramid vertex figure.

H3 633-1101.png

Runcicantellated hexagonal tiling honeycomb

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
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}, CDel node 1.pngCDel 6.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.png has truncated tetrahedron, hexagonal prism, and rhombitrihexagonal tiling cells, with an isosceles-trapezoidal pyramid vertex figure.

H3 633-1011.png

Omnitruncated hexagonal tiling honeycomb

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.svg
Faces square {4}
hexagon {6}
dodecagon {12}
Vertex figure Omnitruncated order-3 hexagonal tiling honeycomb verf.png
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}, 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 an irregular tetrahedron vertex figure.

H3 633-1111.png

See also

Related Research Articles

<span class="mw-page-title-main">Cubic honeycomb</span> Only regular space-filling tessellation of the cube

The cubic honeycomb or cubic cellulation is the only proper regular space-filling tessellation in Euclidean 3-space made up of cubic cells. It has 4 cubes around every edge, and 8 cubes around each vertex. Its vertex figure is a regular octahedron. It is a self-dual tessellation with Schläfli symbol {4,3,4}. John Horton Conway called this honeycomb a cubille.

<span class="mw-page-title-main">Order-5 cubic honeycomb</span> Regular tiling of hyperbolic 3-space

In hyperbolic geometry, the order-5 cubic honeycomb is one of four compact regular space-filling tessellations in hyperbolic 3-space. With Schläfli symbol {4,3,5}, it has five cubes {4,3} around each edge, and 20 cubes around each vertex. It is dual with the order-4 dodecahedral honeycomb.

<span class="mw-page-title-main">Icosahedral honeycomb</span> Regular tiling of hyperbolic 3-space

In geometry, the icosahedral honeycomb is one of four compact, regular, space-filling tessellations 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.

In geometry, a quasiregular polyhedron is a uniform polyhedron that has exactly two kinds of regular faces, which alternate around each vertex. They are vertex-transitive and edge-transitive, hence a step closer to regular polyhedra than the semiregular, which are merely vertex-transitive.

<span class="mw-page-title-main">Uniform honeycombs in hyperbolic space</span> Tiling of hyperbolic 3-space by uniform polyhedra

In hyperbolic geometry, a uniform honeycomb in hyperbolic space is a uniform tessellation of uniform polyhedral cells. In 3-dimensional hyperbolic space there are nine Coxeter group families of compact convex uniform honeycombs, generated as Wythoff constructions, and represented by permutations of rings of the Coxeter diagrams for each family.

In geometry, uniform honeycombs in hyperbolic space are tessellations of convex uniform polyhedron cells. In 3-dimensional hyperbolic space there are 23 Coxeter group families of paracompact uniform honeycombs, generated as Wythoff constructions, and represented by ring permutations of the Coxeter diagrams for each family. These families can produce uniform honeycombs with infinite or unbounded facets or vertex figure, including ideal vertices at infinity, similar to the hyperbolic uniform tilings in 2-dimensions.

<span class="mw-page-title-main">Order-6 tetrahedral honeycomb</span>

In hyperbolic 3-space, the order-6 tetrahedral honeycomb is a paracompact regular space-filling tessellation. It is paracompact because it has vertex figures composed of an infinite number of faces, and has all vertices as ideal points at infinity. With Schläfli symbol {3,3,6}, the order-6 tetrahedral honeycomb has six ideal tetrahedra around each edge. All vertices are ideal, with infinitely many tetrahedra existing around each vertex in a triangular tiling vertex figure.

<span class="mw-page-title-main">Order-4 hexagonal tiling honeycomb</span>

In the field of hyperbolic geometry, the order-4 hexagonal tiling honeycomb arises as 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 flat plane in hyperbolic space that approaches a single ideal point at infinity.

<span class="mw-page-title-main">Order-6 cubic honeycomb</span>

The order-6 cubic honeycomb is a paracompact regular space-filling tessellation in hyperbolic 3-space. It is paracompact because it has vertex figures composed of an infinite number of facets, with all vertices as ideal points at infinity. With Schläfli symbol {4,3,6}, the honeycomb has six ideal cubes meeting along each edge. Its vertex figure is an infinite triangular tiling. Its dual is the order-4 hexagonal tiling honeycomb.

<span class="mw-page-title-main">Order-6 dodecahedral honeycomb</span> Regular geometrical object in hyperbolic space

The order-6 dodecahedral honeycomb is one of 11 paracompact regular honeycombs in hyperbolic 3-space. It is paracompact because it has vertex figures composed of an infinite number of faces, with all vertices as ideal points at infinity. It has Schläfli symbol {5,3,6}, with six ideal dodecahedral cells surrounding each edge of the honeycomb. Each vertex is ideal, and surrounded by infinitely many dodecahedra. The honeycomb has a triangular tiling vertex figure.

<span class="mw-page-title-main">Order-5 hexagonal tiling honeycomb</span>

In the field of hyperbolic geometry, the order-5 hexagonal tiling honeycomb arises as 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 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.

<span class="mw-page-title-main">Order-6 hexagonal tiling honeycomb</span>

In the field of hyperbolic geometry, the order-6 hexagonal tiling honeycomb is one of 11 regular paracompact honeycombs in 3-dimensional hyperbolic space. It is paracompact because it has cells with an infinite number of faces. Each cell is a hexagonal tiling whose vertices lie on a horosphere: a flat plane in hyperbolic space that approaches a single ideal point at infinity.

<span class="mw-page-title-main">Triangular tiling honeycomb</span>

The triangular tiling honeycomb is one of 11 paracompact regular space-filling tessellations in hyperbolic 3-space. It is called paracompact because it has infinite cells and vertex figures, with all vertices as ideal points at infinity. It has Schläfli symbol {3,6,3}, being composed of triangular tiling cells. Each edge of the honeycomb is surrounded by three cells, and each vertex is ideal with infinitely many cells meeting there. Its vertex figure is a hexagonal tiling.

<span class="mw-page-title-main">Square tiling honeycomb</span>

In the geometry of hyperbolic 3-space, the square tiling honeycomb is one of 11 paracompact regular honeycombs. It is called paracompact because it has infinite cells, whose vertices exist on horospheres and converge to a single ideal point at infinity. Given by Schläfli symbol {4,4,3}, it has three square tilings, {4,4}, around each edge, and six square tilings around each vertex, in a cubic {4,3} vertex figure.

<span class="mw-page-title-main">Order-4 square tiling honeycomb</span>

In the geometry of hyperbolic 3-space, the order-4 square tiling honeycomb is one of 11 paracompact regular honeycombs. It is paracompact because it has infinite cells and vertex figures, with all vertices as ideal points at infinity. Given by Schläfli symbol {4,4,4}, it has four square tilings around each edge, and infinite square tilings around each vertex in a square tiling vertex figure.

<span class="mw-page-title-main">Order-4 octahedral honeycomb</span>

The order-4 octahedral honeycomb is a regular paracompact honeycomb in hyperbolic 3-space. It is paracompact because it has infinite vertex figures, with all vertices as ideal points at infinity. Given by Schläfli symbol {3,4,4}, it has four ideal octahedra around each edge, and infinite octahedra around each vertex in a square tiling vertex figure.

In three-dimensional hyperbolic geometry, the alternated hexagonal tiling honeycomb, h{6,3,3}, or , is a semiregular tessellation with tetrahedron and triangular tiling cells arranged in an octahedron vertex figure. It is named after its construction, as an alteration of a hexagonal tiling honeycomb.

<span class="mw-page-title-main">Order-3-7 hexagonal honeycomb</span>

In the geometry of hyperbolic 3-space, the order-3-7 hexagonal honeycomb or a regular space-filling tessellation with Schläfli symbol {6,3,7}.

In the geometry of hyperbolic 3-space, the octahedron-hexagonal tiling honeycomb is a paracompact uniform honeycomb, constructed from octahedron, hexagonal tiling, and trihexagonal tiling cells, in a rhombicuboctahedron vertex figure. It has a single-ring Coxeter diagram, , and is named by its two regular cells.

In the geometry of hyperbolic 3-space, the tetrahedral-square tiling honeycomb is a paracompact uniform honeycomb, constructed from tetrahedron, cuboctahedron and square tiling cells, in a rhombicuboctahedron vertex figure. It has a single-ring Coxeter diagram, , and is named by its two regular cells.

References

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