Order-6 tetrahedral honeycomb

Order-6 tetrahedral honeycomb
Perspective projection view within Poincaré disk model
Type	Hyperbolic regular honeycomb Paracompact uniform honeycomb
Schläfli symbols	{3,3,6} {3,3^[3]}
Coxeter diagrams	↔
Cells	{3,3}
Faces	Triangle {3}
Edge figure	Hexagon {6}
Vertex figure	Triangular tiling {3,6}
Dual	Hexagonal tiling honeycomb, {6,3,3}
Coxeter groups	${\bar {V}}_{3}$ , [6,3,3] ${\bar {P}}_{3}$ , [3,3^[3]]
Properties	Regular, quasiregular

In the geometry of hyperbolic 3-space, the order-6 tetrahedral honeycomb is a paracompact regular space-filling tessellation (or honeycomb). It is called paracompact because it has infinite vertex figures, with all vertices as ideal points at infinity. With Schläfli symbol {3,3,6}. It has six tetrahedra {3,3} around each edge. All vertices are ideal vertices with infinitely many tetrahedra existing around each ideal vertex in an triangular tiling vertex arrangement.^[1]

A 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 in any number of dimensions.

Honeycombs are usually constructed in ordinary 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

Subgroup relations

It has a second construction as a uniform honeycomb, Schläfli symbol {3,3^[3]}, with alternating types or colors of tetrahedral cells. In Coxeter notation the half symmetry is [3,3,6,1⁺] ↔ [3,((3,3,3))] or [3,3^[3]]: ↔ .

Related polytopes and honeycombs

It is similar to the 2-dimensional hyperbolic tiling, infinite-order triangular tiling, {3,∞}, having all triangle faces, and all ideal vertices.

Related regular honeycombs

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

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}

633 honeycombs

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

[6,3,3] family honeycombs
{6,3,3}	r{6,3,3}	t{6,3,3}	rr{6,3,3}	t_0,3{6,3,3}	tr{6,3,3}	t_0,1,3{6,3,3}	t_0,1,2,3{6,3,3}


{3,3,6}	r{3,3,6}	t{3,3,6}	rr{3,3,6}	2t{3,3,6}	tr{3,3,6}	t_0,1,3{3,3,6}	t_0,1,2,3{3,3,6}

Tetrahedral cell honeycombs

It a 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}	{3,3,4}	{3,3,5}	{3,3,6}	{3,3,7}	{3,3,8}	... {3,3,∞}
Image
Vertex figure	{3,3}	{3,4}	{3,5}	{3,6}	{3,7}	{3,8}	{3,∞}

Triangular tiling vertex figures

It a 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} {3,3^[3]}	{4,3,6} {4,3^[3]}	{5,3,6} {5,3^[3]}	{6,3,6} {6,3^[3]}	{7,3,6} {7,3^[3]}	{8,3,6} {8,3^[3]}	... {∞,3,6} {∞,3^[3]}

Image
Cells	{3,3}	{4,3}	{5,3}	{6,3}	{7,3}	{8,3}	{∞,3}

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	↔
Cells	{3,4} {3,6}
Vertex figure	Hexagonal prism { }×{6}
Coxeter groups	${\bar {V}}_{3}$ , [6,3,3] ${\bar {P}}_{3}$ , [3,3^[3]]
Properties	Vertex-transitive, edge-transitive

The rectified order-6 tetrahedral honeycomb, t₁{3,3,6} has octahedral and triangular tiling cells connected in a hexagonal prism vertex figure.

Perspective projection view within Poincaré disk model

r{p,3,6}
Space	H³
Form	Paracompact				Noncompact
Name	r{3,3,6}	r{4,3,6}	r{5,3,6}	r{6,3,6}	r{7,3,6}	... r{∞,3,6}
Image
Cells {3,6}	r{3,3}	r{4,3}	r{6,3}	r{6,3}	r{∞,3}	r{∞,3}

Truncated order-6 tetrahedral honeycomb

Truncated order-6 tetrahedral honeycomb
Type	Paracompact uniform honeycomb
Schläfli symbols	t{3,3,6} or t_0,1{3,3,6}
Coxeter diagrams	↔
Cells	t{3,3} {3,6}
Vertex figure	Hexagonal pyramid { }v{6}
Coxeter groups	${\bar {V}}_{3}$ , [6,3,3] ${\bar {P}}_{3}$ , [3,3^[3]]
Properties	Vertex-transitive

The truncated order-6 tetrahedral honeycomb, t_0,1{3,3,6} has truncated tetrahedra and triangular tiling cells connected in a hexagonal prism vertex figure.

Bitruncated order-6 tetrahedral honeycomb

Further information: 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 t_0,2{3,3,6}
Coxeter diagrams	↔
Cells	r{3,3} r{3,6} {}x{6}
Vertex figure	tetrahedron
Coxeter groups	${\bar {V}}_{3}$ , [6,3,3] ${\bar {P}}_{3}$ , [3,3^[3]]
Properties	Vertex-transitive

The cantellated order-6 tetrahedral honeycomb, t_0,2{3,3,6} has cuboctahedron and trihexagonal tiling cells connected in a tetrahedron vertex figure.

Cantitruncated order-6 tetrahedral honeycomb

Cantitruncated order-6 tetrahedral honeycomb
Type	Paracompact uniform honeycomb
Schläfli symbols	tr{3,3,6} or t_0,1,2{3,3,6}
Coxeter diagrams	↔
Cells	tr{3,3} r{3,6} {}x{6}
Vertex figure	tetrahedron
Coxeter groups	${\bar {V}}_{3}$ , [6,3,3] ${\bar {P}}_{3}$ , [3,3^[3]]
Properties	Vertex-transitive

The cantitruncated order-6 tetrahedral honeycomb, t_0,1,2{3,3,6} has truncated cuboctahedron and trihexagonal tiling cells connected in an octahedron vertex figure.

References

↑ 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 (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, (2015) Chapter 13: Hyperbolic Coxeter groups

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