Triangular tiling honeycomb

Triangular tiling honeycomb
Type	Hyperbolic regular honeycomb Paracompact uniform honeycomb
Schläfli symbol	{3,6,3} h{6,3,6} h{6,3^[3]} ↔ {3^[3,3]}
Coxeter-Dynkin diagrams	↔ ↔ ↔
Cells	{3,6}
Faces	triangle {3}
Edge figure	triangle {3}
Vertex figure	{6,3}
Dual	Self-dual
Coxeter groups	Y3, [3,6,3] ${\bar {PP}}_{3}$, [3^[3,3]]
Properties	Regular

The triangular tiling honeycomb is one of 11 paracompact regular space-filling tessellations (or honeycombs) 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.

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

Subgroups of [3,6,3] and [6,3,6]

It has two lower reflective symmetry constructions, as an alternated order-6 hexagonal tiling honeycomb, ↔ , and as from , which alternates 3 types (colors) of triangular tilings around every edge. In Coxeter notation, the removal of the 3rd and 4th mirrors, [3,6,3^*] creates a new Coxeter group [3^[3,3]], , subgroup index 6. The fundamental domain is 6 times larger. By Coxeter diagram there are 3 copies of the first original mirror in the new fundamental domain: ↔ .

Images

It is similar to the 2D hyperbolic infinite-order apeirogonal tiling, {∞,∞} with infinite apeirogonal faces and all vertices are on the ideal surface.

Related honeycombs

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.

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}

There are nine uniform honeycombs in the [3,6,3] Coxeter group family, including this regular form as well as the bitruncated form, t_1,2{3,6,3}, with all truncated hexagonal tiling facets.

[3,6,3] family honeycombs

{3,6,3}	r{3,6,3}	t{3,6,3}	rr{3,6,3}	t0,3{3,6,3}	2t{3,6,3}	tr{3,6,3}	t0,1,3{3,6,3}	t0,1,2,3{3,6,3}

{3,p,3} polytopes
Space	S^3		H^3
Form	Finite		Compact	Paracompact	Noncompact
{3,p,3}	{3,3,3}	{3,4,3}	{3,5,3}	{3,6,3}	{3,7,3}	{3,8,3}	... {3,∞,3}
Image
Cells	{3,3}	{4,3}	{3,5}	{3,6}	{3,7}	{3,8}	{3,∞}
Vertex figure	{3,3}	{4,3}	{5,3}	{6,3}	{7,3}	{8,3}	{∞,3}

Rectified triangular tiling honeycomb

Rectified triangular tiling honeycomb
Type	Paracompact uniform honeycomb
Schläfli symbol	r{3,6,3} h2{6,3,6}
Coxeter diagram	↔ ↔ ↔
Cells	r{3,6} {6,3}
Faces	triangle {3} Hexagon {6}
Vertex figure	Triangular prism
Coxeter group	Y3, [3,6,3]
Properties	Vertex-transitive

The rectified triangular tiling honeycomb, , has alternating trihexagonal tiling and hexagonal tiling cells, with a triangular prism vertex figure.

Symmetry

A lower symmetry of this honeycomb can be constructed as a cantic order-6 hexagonal tiling honeycomb, ↔ . A second lower construction, index is ↔ .

Truncated triangular tiling honeycomb

The truncated triangular tiling honeycomb,, is the same as the hexagonal tiling honeycomb, .

Bitruncated triangular tiling honeycomb

Bitruncated triangular tiling honeycomb
Type	Paracompact uniform honeycomb
Schläfli symbol	2t{3,6,3}
Coxeter diagram
Cells	t{6,3}
Faces	Triangle {3} Dodecagon {12}
Vertex figure	Tetrahedron
Coxeter group	Y3×2, [[3,6,3]]
Properties	Vertex-transitive, edge-transitive, cell-transitive

The bitruncated triangular tiling honeycomb, , has truncated hexagonal tiling cells, with a tetrahedral vertex figure.

Cantellated triangular tiling honeycomb

Cantellated triangular tiling honeycomb
Type	Paracompact uniform honeycomb
Schläfli symbol	rr{3,6,3} or t0,2{3,6,3} s2{3,6,3}
Coxeter diagram
Cells	rr{6,3} r{6,3} {}×{3}
Faces	Triangle {3} Square {4} Hexagon {6}
Vertex figure	triangular prism
Coxeter group	Y3, [3,6,3]
Properties	Vertex-transitive

The cantellated triangular tiling honeycomb, , has truncated hexagonal tiling and triangular prism cells, with a triangular prism vertex figure.

Symmetry

It can also be constructed as a half-symmetry form, cantic snub triangular tiling honeycomb, , symmetry [3⁺,6,3].

Cantitruncated triangular tiling honeycomb

Cantitruncated triangular tiling honeycomb
Type	Paracompact uniform honeycomb
Schläfli symbol	tr{3,6,3} or t0,1,2{3,6,3}
Coxeter diagram
Cells	tr{6,3} t{6,3} {}×{6}
Faces	Triangle {3} Square {4} Hexagon {6} Dodecagon {12}
Vertex figure	tetrahedron
Coxeter group	Y3, [3,6,3]
Properties	Vertex-transitive

The cantitruncated triangular tiling honeycomb, , has rhombitrihexagonal tiling and hexagonal prism cells, with a tetrahedron vertex figure.

Runcinated triangular tiling honeycomb

Runcinated triangular tiling honeycomb
Type	Paracompact uniform honeycomb
Schläfli symbol	t0,3{3,6,3}
Coxeter diagram
Cells	{3,6} {}×{3}
Faces	Triangle {3} Square {4}
Vertex figure	hexagonal antiprism
Coxeter group	2×Y3, [[3,6,3]]
Properties	Vertex-transitive, edge-transitive

The runcinated triangular tiling honeycomb, , has hexagonal tiling and triangular prism cells, with a hexagonal antiprism vertex figure.

Runcitruncated triangular tiling honeycomb

Runcitruncated triangular tiling honeycomb
Type	Paracompact uniform honeycomb
Schläfli symbols	t0,1,3{3,6,3} s2,3{3,6,3}
Coxeter diagrams
Cells	t{3,6} rr{3,6} {}×{3} {}×{6}
Faces	Triangle {3} Square {4} Hexagon {6}
Vertex figure	quadrilateral pyramid
Coxeter group	Y3, [3,6,3]
Properties	Vertex-transitive

The runcitruncated triangular tiling honeycomb, , has hexagonal tiling, rhombitrihexagonal tiling, triangular prism, and dodecagonal prism cells.

Omnitruncated triangular tiling honeycomb

Omnitruncated triangular tiling honeycomb
Type	Paracompact uniform honeycomb
Schläfli symbol	t0,1,2,3{3,6,3}
Coxeter diagram
Cells	tr{3,6} {}×{12}
Faces	Square {4} Hexagon {6} Dodecagon {12}
Vertex figure	Phyllic disphenoid
Coxeter group	2×Y3, [[3,6,3]]
Properties	Vertex-transitive, edge-transitive

The omnitruncated triangular tiling honeycomb, , has truncated trihexagonal tiling and hexagonal prism cells, with a tetrahedron vertex figure.

Runcisnub triangular tiling honeycomb

Runcisnub triangular tiling honeycomb
Type	Paracompact scaliform honeycomb
Schläfli symbol	s3{3,6,3}
Coxeter diagram
Cells	r{6,3} {}x{3} {3,6} tricup
Faces	Triangle {3} Square {4} hexagon {6}
Vertex figure
Coxeter group	Y3, [3^+,6,3]
Properties	Vertex-transitive

The runcisnub triangular tiling honeycomb, , has trihexagonal tiling, triangular tiling, triangular prism, and triangular cupola cells. It is vertex-transitive, but not uniform, since it contains Johnson solid triangular cupola cells.

See also

• Convex uniform honeycombs in hyperbolic space

References

• 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