Order-6 tetrahedral honeycomb
File:H3 336 CC center.png Perspective projection viewPoincaré disk model
Type
Hyperbolic regular honeycomb Paracompact uniform honeycomb
Schläfli symbols {3,3,6}[3] }
Coxeter diagrams File:CDel node 1.png File:CDel 3.png File:CDel node.png File:CDel 3.png File:CDel node.png File:CDel 6.png File:CDel node.png File:CDel node 1.png File:CDel 3.png File:CDel node.png File:CDel 3.png File:CDel node.png File:CDel 6.png File:CDel node h0.png File:CDel node 1.png File:CDel 3.png File:CDel node.png File:CDel split1.png File:CDel branch.png
Cells
{3,3} File:Uniform polyhedron-33-t0.png
Faces
triangle {3}
Edge figure hexagon {6}
Vertex figure File:Uniform tiling 63-t2.png File:Uniform tiling 333-t1.svg triangular tiling
Dual Hexagonal tiling honeycomb
Coxeter groups V ‾ 3 P ‾ 3 [3] ]
Properties
Regular, quasiregular
In hyperbolic 3-space , the order-6 tetrahedral honeycomb is a paracompact regular space-filling tessellation (or honeycomb ). 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 .[ 1] geometric honeycomb is a space-filling of polyhedral or higher-dimensional cells , so that there are no gaps. It is an example of the more general mathematical tiling or tessellation Euclidean ("flat") space, like the convex uniform honeycombs . They may also be constructed in non-Euclidean spaces , such as hyperbolic uniform honeycombs . Any finite uniform polytope can be projected to its circumsphere to form a uniform honeycomb in spherical space.
Symmetry constructions File:Hyperbolic subgroup tree 336-direct.png Subgroup relations The order-6 tetrahedral honeycomb has a second construction as a uniform honeycomb, with Schläfli symbol {3,3[3] }. This construction contains alternating types, or colors, of tetrahedral cells. In Coxeter notation , this half symmetry is represented as [3,3,6,1+ ] ↔ [3,((3,3,3))], or [3,3[3] ]: File:CDel node c1.png File:CDel 3.png File:CDel node c2.png File:CDel 3.png File:CDel node c3.png File:CDel 6.png File:CDel node h0.png File:CDel node c1.png File:CDel 3.png File:CDel node c2.png File:CDel split1.png File:CDel branch c3.png
Related polytopes and honeycombs The order-6 tetrahedral honeycomb is analogous to the two-dimensional infinite-order triangular tiling , {3,∞}. Both tessellations are regular, and only contain triangles and ideal vertices.
File:Infinite-order triangular tiling.svg The order-6 tetrahedral honeycomb is also a regular hyperbolic honeycomb in 3-space, and one of 11 which are paracompact.
11 paracompact regular honeycombs
File:H3 633 FC boundary.png {6,3,3}
File:H3 634 FC boundary.png {6,3,4}
File:H3 635 FC boundary.png {6,3,5}
File:H3 636 FC boundary.png {6,3,6}
File:H3 443 FC boundary.png {4,4,3}
File:H3 444 FC boundary.png {4,4,4}
File:H3 336 CC center.png {3,3,6}
File:H3 436 CC center.png {4,3,6}
File:H3 536 CC center.png {5,3,6}
File:H3 363 FC boundary.png {3,6,3}
File:H3 344 CC center.png {3,4,4}
This honeycomb is one of 15 uniform paracompact honeycombs in the [6,3,3] Coxeter group, along with its dual, the hexagonal tiling 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}
File:H3 633 FC boundary.png File:H3 633 boundary 0100.png File:H3 633-1100.png File:H3 633-1010.png File:H3 633-1001.png File:H3 633-1110.png File:H3 633-1101.png File:H3 633-1111.png
File:H3 336 CC center.png File:H3 336 CC center 0100.png File:H3 633-0011.png File:H3 633-0101.png File:H3 633-0110.png File:H3 633-0111.png File: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}
The order-6 tetrahedral honeycomb is part of a sequence of regular polychora and honeycombs with tetrahedral cells .
{3,3,p} polytopes
Space
S3
H3
Form
Finite
Paracompact
Noncompact
Name
{3,3,3} File:CDel node 1.png File:CDel 3.png File:CDel node.png File:CDel 3.png File:CDel node.png File:CDel 3.png File:CDel node.png {3,3,4} File:CDel node 1.png File:CDel 3.png File:CDel node.png File:CDel 3.png File:CDel node.png File:CDel 4.png File:CDel node.png File:CDel node 1.png File:CDel 3.png File:CDel node.png File:CDel split1.png File:CDel nodes.png {3,3,5} File:CDel node 1.png File:CDel 3.png File:CDel node.png File:CDel 3.png File:CDel node.png File:CDel 5.png File:CDel node.png {3,3,6} File:CDel node 1.png File:CDel 3.png File:CDel node.png File:CDel 3.png File:CDel node.png File:CDel 6.png File:CDel node.png File:CDel node 1.png File:CDel 3.png File:CDel node.png File:CDel split1.png File:CDel branch.png {3,3,7} File:CDel node 1.png File:CDel 3.png File:CDel node.png File:CDel 3.png File:CDel node.png File:CDel 7.png File:CDel node.png {3,3,8} File:CDel node 1.png File:CDel 3.png File:CDel node.png File:CDel 3.png File:CDel node.png File:CDel 8.png File:CDel node.png File:CDel node 1.png File:CDel 3.png File:CDel node.png File:CDel split1.png File:CDel branch.png File:CDel label4.png ... {3,3,∞} File:CDel node 1.png File:CDel 3.png File:CDel node.png File:CDel 3.png File:CDel node.png File:CDel infin.png File:CDel node.png File:CDel node 1.png File:CDel 3.png File:CDel node.png File:CDel split1.png File:CDel branch.png File:CDel labelinfin.png
Image
File:Stereographic polytope 5cell.png File:Stereographic polytope 16cell.png File:Stereographic polytope 600cell.png File:H3 336 CC center.png File:Hyperbolic honeycomb 3-3-7 poincare cc.png File:Hyperbolic honeycomb 3-3-8 poincare cc.png File:Hyperbolic honeycomb 3-3-i poincare cc.png
Vertex
File:5-cell verf.svg {3,3} File:CDel node 1.png File:CDel 3.png File:CDel node.png File:CDel 3.png File:CDel node.png File:16-cell verf.svg {3,4} File:CDel node 1.png File:CDel 3.png File:CDel node.png File:CDel 4.png File:CDel node.png File:CDel node 1.png File:CDel split1.png File:CDel nodes.png File:600-cell verf.svg {3,5} File:CDel node 1.png File:CDel 3.png File:CDel node.png File:CDel 5.png File:CDel node.png File:Uniform tiling 63-t2.svg {3,6} File:CDel node 1.png File:CDel 3.png File:CDel node.png File:CDel 6.png File:CDel node.png File:CDel node 1.png File:CDel split1.png File:CDel branch.png File:Order-7 triangular tiling.svg {3,7} File:CDel node 1.png File:CDel 3.png File:CDel node.png File:CDel 7.png File:CDel node.png File:H2-8-3-primal.svg {3,8} File:CDel node 1.png File:CDel 3.png File:CDel node.png File:CDel 8.png File:CDel node.png File:CDel node 1.png File:CDel split1.png File:CDel branch.png File:CDel label4.png File:H2 tiling 23i-4.png {3,∞} File:CDel node 1.png File:CDel 3.png File:CDel node.png File:CDel infin.png File:CDel node.png File:CDel node 1.png File:CDel split1.png File:CDel branch.png File:CDel labelinfin.png
It is also part of a sequence of honeycombs with triangular tiling vertex figures .
Hyperbolic uniform honeycombs : {p ,3,6} and {p ,3[3] }
Form
Paracompact
Noncompact
Name
{3,3,6} [3] }
{4,3,6} [3] }
{5,3,6} [3] }
{6,3,6} [3] }
{7,3,6} [3] }
{8,3,6} [3] }
... {∞,3,6} [3] }
File:CDel node 1.png File:CDel p.png File:CDel node.png File:CDel 3.png File:CDel node.png File:CDel 6.png File:CDel node.png File:CDel node 1.png File:CDel p.png File:CDel node.png File:CDel split1.png File:CDel branch.png File:CDel node 1.png File:CDel 3.png File:CDel node.png File:CDel 3.png File:CDel node.png File:CDel 6.png File:CDel node.png File:CDel node 1.png File:CDel 3.png File:CDel node.png File:CDel split1.png File:CDel branch.png File:CDel node 1.png File:CDel 4.png File:CDel node.png File:CDel 3.png File:CDel node.png File:CDel 6.png File:CDel node.png File:CDel node 1.png File:CDel 4.png File:CDel node.png File:CDel split1.png File:CDel branch.png File:CDel node 1.png File:CDel 5.png File:CDel node.png File:CDel 3.png File:CDel node.png File:CDel 6.png File:CDel node.png File:CDel node 1.png File:CDel 5.png File:CDel node.png File:CDel split1.png File:CDel branch.png File:CDel node 1.png File:CDel 6.png File:CDel node.png File:CDel 3.png File:CDel node.png File:CDel 6.png File:CDel node.png File:CDel node 1.png File:CDel 6.png File:CDel node.png File:CDel split1.png File:CDel branch.png File:CDel node 1.png File:CDel 7.png File:CDel node.png File:CDel 3.png File:CDel node.png File:CDel 6.png File:CDel node.png File:CDel node 1.png File:CDel 7.png File:CDel node.png File:CDel split1.png File:CDel branch.png File:CDel node 1.png File:CDel 8.png File:CDel node.png File:CDel 3.png File:CDel node.png File:CDel 6.png File:CDel node.png File:CDel node 1.png File:CDel 8.png File:CDel node.png File:CDel split1.png File:CDel branch.png File:CDel node 1.png File:CDel infin.png File:CDel node.png File:CDel 3.png File:CDel node.png File:CDel 6.png File:CDel node.png File:CDel node 1.png File:CDel infin.png File:CDel node.png File:CDel split1.png File:CDel branch.png
Image
File:H3 336 CC center.png File:H3 436 CC center.png File:H3 536 CC center.png File:H3 636 FC boundary.png File:Hyperbolic honeycomb 7-3-6 poincare.png File:Hyperbolic honeycomb 8-3-6 poincare.png File:Hyperbolic honeycomb i-3-6 poincare.png
Cells
File:Tetrahedron.png {3,3} File:CDel node 1.png File:CDel 3.png File:CDel node.png File:CDel 3.png File:CDel node.png File:Hexahedron.png {4,3} File:CDel node 1.png File:CDel 4.png File:CDel node.png File:CDel 3.png File:CDel node.png File:Dodecahedron.png {5,3} File:CDel node 1.png File:CDel 5.png File:CDel node.png File:CDel 3.png File:CDel node.png File:Uniform tiling 63-t0.svg {6,3} File:CDel node 1.png File:CDel 6.png File:CDel node.png File:CDel 3.png File:CDel node.png File:Heptagonal tiling.svg {7,3} File:CDel node 1.png File:CDel 7.png File:CDel node.png File:CDel 3.png File:CDel node.png File:H2-8-3-dual.svg {8,3} File:CDel node 1.png File:CDel 8.png File:CDel node.png File:CDel 3.png File:CDel node.png File:H2-I-3-dual.svg {∞,3} File:CDel node 1.png File:CDel infin.png File:CDel node.png File:CDel 3.png File:CDel node.png
Rectified order-6 tetrahedral honeycomb
Rectified order-6 tetrahedral honeycomb
Type
Paracompact uniform honeycomb Semiregular honeycomb
Schläfli symbols r{3,3,6} or t1 {3,3,6}
Coxeter diagrams File:CDel node.png File:CDel 3.png File:CDel node 1.png File:CDel 3.png File:CDel node.png File:CDel 6.png File:CDel node.png File:CDel node.png File:CDel 3.png File:CDel node 1.png File:CDel 3.png File:CDel node.png File:CDel 6.png File:CDel node h0.png File:CDel node.png File:CDel 3.png File:CDel node 1.png File:CDel split1.png File:CDel branch.png
Cells
r{3,3} File:Uniform polyhedron-33-t1.svg {3,6} File:Uniform tiling 63-t2.png
Faces
triangle {3}
Vertex figure File:Rectified order-6 tetrahedral honeycomb verf.png hexagonal prism
Coxeter groups V ‾ 3 P ‾ 3 [3] ]
Properties
Vertex-transitive, edge-transitive
The rectified order-6 tetrahedral honeycomb , t1 {3,3,6} has octahedral and triangular tiling cells arranged in a hexagonal prism vertex figure .
File:H3 336 CC center 0100.png File:Hyperbolic rectified order-6 tetrahedral honeycomb.png Perspective projection view within Poincaré disk model
r{p,3,6}
Space
H3
Form
Paracompact
Noncompact
Name
r{3,3,6} File:CDel node.png File:CDel 3.png File:CDel node 1.png File:CDel 3.png File:CDel node.png File:CDel 6.png File:CDel node.png r{4,3,6} File:CDel node.png File:CDel 4.png File:CDel node 1.png File:CDel 3.png File:CDel node.png File:CDel 6.png File:CDel node.png r{5,3,6} File:CDel node.png File:CDel 5.png File:CDel node 1.png File:CDel 3.png File:CDel node.png File:CDel 6.png File:CDel node.png r{6,3,6} File:CDel node.png File:CDel 6.png File:CDel node 1.png File:CDel 3.png File:CDel node.png File:CDel 6.png File:CDel node.png r{7,3,6}File:CDel node.png File:CDel 7.png File:CDel node 1.png File:CDel 3.png File:CDel node.png File:CDel 6.png File:CDel node.png
... r{∞,3,6}File:CDel node.png File:CDel infin.png File:CDel node 1.png File:CDel 3.png File:CDel node.png File:CDel 6.png File:CDel node.png
Image
File:H3 336 CC center 0100.png File:H3 436 CC center 0100.png File:H3 536 CC center 0100.png File:H3 636 boundary 0100.png
CellsFile:Uniform tiling 63-t2.svg {3,6} File:CDel node 1.png File:CDel 3.png File:CDel node.png File:CDel 6.png File:CDel node.png
File:Uniform polyhedron-33-t1.svg r{3,3} File:CDel node.png File:CDel 3.png File:CDel node 1.png File:CDel 3.png File:CDel node.png File:Cuboctahedron.png r{4,3} File:CDel node.png File:CDel 4.png File:CDel node 1.png File:CDel 3.png File:CDel node.png File:Icosidodecahedron.png r{5,3} File:CDel node.png File:CDel 5.png File:CDel node 1.png File:CDel 3.png File:CDel node.png File:Uniform tiling 63-t1.svg r{6,3} File:CDel node.png File:CDel 6.png File:CDel node 1.png File:CDel 3.png File:CDel node.png File:Triheptagonal tiling.svg r{7,3} File:CDel node.png File:CDel 7.png File:CDel node 1.png File:CDel 3.png File:CDel node.png File:H2 tiling 23i-2.png File:CDel node.png File:CDel infin.png File:CDel node 1.png File:CDel 3.png File:CDel node.png
Truncated order-6 tetrahedral honeycomb
Truncated order-6 tetrahedral honeycomb
Type
Paracompact uniform honeycomb
Schläfli symbols t{3,3,6} or t0,1 {3,3,6}
Coxeter diagrams File:CDel node 1.png File:CDel 3.png File:CDel node 1.png File:CDel 3.png File:CDel node.png File:CDel 6.png File:CDel node.png File:CDel node 1.png File:CDel 3.png File:CDel node 1.png File:CDel 3.png File:CDel node.png File:CDel 6.png File:CDel node h0.png File:CDel node 1.png File:CDel 3.png File:CDel node 1.png File:CDel split1.png File:CDel branch.png
Cells
t{3,3} File:Uniform polyhedron-33-t01.png {3,6} File:Uniform tiling 63-t2.png
Faces
triangle {3}hexagon {6}
Vertex figure File:Truncated order-6 tetrahedral honeycomb verf.png hexagonal pyramid
Coxeter groups V ‾ 3 P ‾ 3 [3] ]
Properties
Vertex-transitive
The truncated order-6 tetrahedral honeycomb , t0,1 {3,3,6} has truncated tetrahedron and triangular tiling cells arranged in a hexagonal pyramid vertex figure .
File:H3 633-0011.png
Bitruncated order-6 tetrahedral honeycomb The bitruncated order-6 tetrahedral honeycomb is equivalent to the bitruncated hexagonal tiling honeycomb .
Cantellated order-6 tetrahedral honeycomb
Cantellated order-6 tetrahedral honeycomb
Type
Paracompact uniform honeycomb
Schläfli symbols rr{3,3,6} or t0,2 {3,3,6}
Coxeter diagrams File:CDel node 1.png File:CDel 3.png File:CDel node.png File:CDel 3.png File:CDel node 1.png File:CDel 6.png File:CDel node.png File:CDel node 1.png File:CDel 3.png File:CDel node.png File:CDel 3.png File:CDel node 1.png File:CDel 6.png File:CDel node h0.png File:CDel node 1.png File:CDel 3.png File:CDel node.png File:CDel split1.png File:CDel branch 11.png
Cells
r{3,3} File:Uniform polyhedron-33-t02.png r{3,6} File:Uniform tiling 63-t1.png {}x{6} File:Hexagonal prism.png
Faces
triangle {3}square {4}hexagon {6}
Vertex figure File:Cantellated order-6 tetrahedral honeycomb verf.png triangular prism
Coxeter groups V ‾ 3 P ‾ 3 [3] ]
Properties
Vertex-transitive
The cantellated order-6 tetrahedral honeycomb , t0,2 {3,3,6} has cuboctahedron , trihexagonal tiling , and hexagonal prism cells arranged in an isosceles triangular prism vertex figure .
File:H3 633-0101.png
Cantitruncated order-6 tetrahedral honeycomb
Cantitruncated order-6 tetrahedral honeycomb
Type
Paracompact uniform honeycomb
Schläfli symbols tr{3,3,6} or t0,1,2 {3,3,6}
Coxeter diagrams File:CDel node 1.png File:CDel 3.png File:CDel node 1.png File:CDel 3.png File:CDel node 1.png File:CDel 6.png File:CDel node.png File:CDel node 1.png File:CDel 3.png File:CDel node 1.png File:CDel 3.png File:CDel node 1.png File:CDel 6.png File:CDel node h0.png File:CDel node 1.png File:CDel 3.png File:CDel node 1.png File:CDel split1.png File:CDel branch 11.png
Cells
tr{3,3} File:Uniform polyhedron-33-t012.png t{3,6} File:Uniform tiling 63-t12.svg {}x{6} File:Hexagonal prism.png
Faces
square {4}hexagon {6}
Vertex figure File:Cantitruncated order-6 tetrahedral honeycomb verf.png mirrored sphenoid
Coxeter groups V ‾ 3 P ‾ 3 [3] ]
Properties
Vertex-transitive
The cantitruncated order-6 tetrahedral honeycomb , t0,1,2 {3,3,6} has truncated octahedron , hexagonal tiling , and hexagonal prism cells connected in a mirrored sphenoid vertex figure .
File:H3 633-0111.png
Runcinated order-6 tetrahedral honeycomb The bitruncated order-6 tetrahedral honeycomb is equivalent to the bitruncated hexagonal tiling honeycomb .
Runcitruncated order-6 tetrahedral honeycomb The runcitruncated order-6 tetrahedral honeycomb is equivalent to the runcicantellated hexagonal tiling honeycomb .
Runcicantellated order-6 tetrahedral honeycomb The runcicantellated order-6 tetrahedral honeycomb is equivalent to the runcitruncated hexagonal tiling honeycomb .
Omnitruncated order-6 tetrahedral honeycomb The omnitruncated order-6 tetrahedral honeycomb is equivalent to the omnitruncated hexagonal tiling honeycomb .
See also References
↑ Coxeter The Beauty of Geometry , 1999, Chapter 10, Table III
Coxeter , Regular Polytopes 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 IIIJeffrey 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, 1966N.W. Johnson: Geometries and Transformations , (2018) Chapter 13: Hyperbolic Coxeter groups 
