Triangular tiling honeycomb explained

bgcolor=#e7dcc3 colspan=2	Triangular tiling honeycomb
bgcolor=#ffffff align=center colspan=2
Type	Hyperbolic regular honeycomb Paracompact uniform honeycomb
Schläfli symbol	h h ↔
Coxeter-Dynkin diagrams	↔ ↔ ↔
Cells
Faces
Edge figure
Vertex figure	hexagonal tiling
Dual	Self-dual
Coxeter groups	\overline{Y} _{3} , [3,6,3] \overline{VP} _{3} , [6,3<sup>[3]] \overline{PP} _{3} , [3<sup>[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, 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.

Symmetry

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<sup>[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: ↔ .

Related Tilings

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

Related honeycombs

The triangular tiling honeycomb is a regular hyperbolic honeycomb in 3-space, and one of eleven paracompact honeycombs.

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, with all truncated hexagonal tiling facets.

The honeycomb is also part of a series of polychora and honeycombs with triangular edge figures.

Rectified triangular tiling honeycomb

bgcolor=#e7dcc3 colspan=2	Rectified triangular tiling honeycomb	-	bgcolor=#ffffff align=center colspan=2	-->
Type	Paracompact uniform honeycomb
width=100 bgcolor=#e7dcc3	Schläfli symbol	r h2
Coxeter diagram	↔ ↔ ↔
Cells
Faces
Vertex figure	triangular prism
Coxeter group	\overline{Y} _{3} , [3,6,3] \overline{VP} _{3} , [6,3<sup>[3]] \overline{PP} _{3} , [3<sup>[3,3]]
Properties	Vertex-transitive, edge-transitive

The rectified triangular tiling honeycomb,, has 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-index construction is ↔ .

Truncated triangular tiling honeycomb

bgcolor=#e7dcc3 colspan=2	Truncated triangular tiling honeycomb	-	bgcolor=#ffffff align=center colspan=2	-->
Type	Paracompact uniform honeycomb
width=100 bgcolor=#e7dcc3	Schläfli symbol	t
Coxeter diagram
Cells
Faces
Vertex figure	tetrahedron
Coxeter group	\overline{Y} _{3} , [3,6,3] \overline{V} _{3} , [3,3,6]
Properties	Regular

The truncated triangular tiling honeycomb,, is a lower-symmetry form of the hexagonal tiling honeycomb, . It contains hexagonal tiling facets with a tetrahedral vertex figure.

Bitruncated triangular tiling honeycomb

bgcolor=#e7dcc3 colspan=2	Bitruncated triangular tiling honeycomb	-	bgcolor=#ffffff align=center colspan=2	-->
Type	Paracompact uniform honeycomb
width=100 bgcolor=#e7dcc3	Schläfli symbol	2t
Coxeter diagram
Cells
Faces
Vertex figure	tetragonal disphenoid
Coxeter group	2 x \overline{Y} _{3} , [[3,6,3]]
Properties	Vertex-transitive, edge-transitive, cell-transitive

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

Cantellated triangular tiling honeycomb

bgcolor=#e7dcc3 colspan=2	Cantellated triangular tiling honeycomb	-	bgcolor=#ffffff align=center colspan=2	-->
Type	Paracompact uniform honeycomb
width=100 bgcolor=#e7dcc3	Schläfli symbol	rr or t0,2 s2
Coxeter diagram
Cells
Faces
Vertex figure	wedge
Coxeter group	\overline{Y} _{3} , [3,6,3]
Properties	Vertex-transitive

The cantellated triangular tiling honeycomb,, has rhombitrihexagonal tiling, trihexagonal tiling, and triangular prism cells, with a wedge vertex figure.

Symmetry

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

Cantitruncated triangular tiling honeycomb

bgcolor=#e7dcc3 colspan=2	Cantitruncated triangular tiling honeycomb	-	bgcolor=#ffffff align=center colspan=2	-->
Type	Paracompact uniform honeycomb
width=100 bgcolor=#e7dcc3	Schläfli symbol	tr or t0,1,2
Coxeter diagram
Cells
Faces
Vertex figure	mirrored sphenoid
Coxeter group	\overline{Y} _{3} , [3,6,3]
Properties	Vertex-transitive

The cantitruncated triangular tiling honeycomb,, has truncated trihexagonal tiling, truncated hexagonal tiling, and triangular prism cells, with a mirrored sphenoid vertex figure.

Runcinated triangular tiling honeycomb

bgcolor=#e7dcc3 colspan=2	Runcinated triangular tiling honeycomb	-	bgcolor=#ffffff align=center colspan=2	-->
Type	Paracompact uniform honeycomb
width=100 bgcolor=#e7dcc3	Schläfli symbol	t0,3
Coxeter diagram
Cells
Faces
Vertex figure	hexagonal antiprism
Coxeter group	2 x \overline{Y} _{3} , [[3,6,3]]
Properties	Vertex-transitive, edge-transitive

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

Runcitruncated triangular tiling honeycomb

bgcolor=#e7dcc3 colspan=2	Runcitruncated triangular tiling honeycomb	-	bgcolor=#ffffff align=center colspan=2	-->
Type	Paracompact uniform honeycomb
width=100 bgcolor=#e7dcc3	Schläfli symbols	t0,1,3 s2,3
Coxeter diagrams
Cells
Faces
Vertex figure	isosceles-trapezoidal pyramid
Coxeter group	\overline{Y} _{3} , [3,6,3]
Properties	Vertex-transitive

The runcitruncated triangular tiling honeycomb,, has hexagonal tiling, rhombitrihexagonal tiling, triangular prism, and hexagonal prism cells, with an isosceles-trapezoidal pyramid vertex figure.

Symmetry

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

Omnitruncated triangular tiling honeycomb

bgcolor=#e7dcc3 colspan=2	Omnitruncated triangular tiling honeycomb	-	bgcolor=#ffffff align=center colspan=2	-->
Type	Paracompact uniform honeycomb
width=100 bgcolor=#e7dcc3	Schläfli symbol	t0,1,2,3
Coxeter diagram
Cells
Faces
Vertex figure	phyllic disphenoid
Coxeter group	2 x \overline{Y} _{3} , [[3,6,3]]
Properties	Vertex-transitive, edge-transitive

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

Runcisnub triangular tiling honeycomb

bgcolor=#e7dcc3 colspan=2	Runcisnub triangular tiling honeycomb	-	bgcolor=#ffffff align=center colspan=2	-->
Type	Paracompact scaliform honeycomb
width=100 bgcolor=#e7dcc3	Schläfli symbol	s3
Coxeter diagram
Cells
Faces
Vertex figure
Coxeter group	\overline{Y} _{3} , [3<sup>+</sup>,6,3]
Properties	Vertex-transitive, non-uniform

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
• Regular tessellations of hyperbolic 3-space
• Paracompact uniform honeycombs

References

• Coxeter, Regular Polytopes, 3rd. ed., Dover Publications, 1973. . (Tables I and II: Regular polytopes and honeycombs, pp. 294–296)
• The Beauty of Geometry: Twelve Essays (1999), Dover Publications,, (Chapter 10, Regular Honeycombs in Hyperbolic Space) Table III
• Jeffrey R. Weeks The Shape of Space, 2nd edition (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, (2018) Chapter 13: Hyperbolic Coxeter groups