Hexagonal tiling honeycomb

Hexagonal tiling honeycomb
Rank	4
Type	Regular, paracompact
Space	Hyperbolic
Notation
Bowers style acronym	Hexah
Coxeter diagram	x6o3o3o ()
Schläfli symbol	{6,3,3}
Elements
Cells	2N hexagonal tilings
Faces	NM hexagons
Edges	2NM
Vertices	NM
Vertex figure	Tetrahedron, edge length √3
Measures (edge length 1)
Circumradius	$i{\sqrt {2}}\approx 1.41421i$
Related polytopes
Army	Hexah
Regiment	Hexah
Dual	Tetrahedral honeycomb
Petrie dual	Petrial hexagonal tiling honeycomb
Abstract & topological properties
Orientable	Yes
Properties
Symmetry	[6,3,3]
Convex	Yes

The hexagonal tiling honeycomb, also known as the order-3 hexagonal tiling honeycomb, is a paracompact regular tiling of 3D hyperbolic space. 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. 3 hexagonal tilings meet at each edge, and 4 meet at each vertex.

It can be seen as a truncated triangular tiling honeycomb or bitruncated order-6 hexagonal tiling honeycomb.

This honeycomb can be alternated into an alternated hexagonal tiling honeycomb, which is uniform.

Representations

The hexagonal tiling honeycomb has the following Coxeter diagrams:

• x6o3o3o () (full symmetry)
• x3x6o3o () (as truncated triangular tiling honeycomb)
• o6x3x6o () (as bitruncated order-6 hexagonal tiling honeycomb)
• o6x3x3x3*b () (half symmetry of bitruncated order-6 hexagonal tiling honeycomb)
• x3x3x3x3*a3*c *b3*d () (quarter symmetry of bitruncated order-6 hexagonal tiling honeycomb)

Related polytopes

o6o3o6o truncations

Name	OBSA	Schläfli symbol	CD diagram	Picture
Order-6 hexagonal tiling honeycomb	hihexah	{6,3,6}
Truncated order-6 hexagonal tiling honeycomb	thihexah	t{6,3,6}
Rectified order-6 hexagonal tiling honeycomb	rihihexah	r{6,3,6}
Dishexagonal tiling honeycomb = Hexagonal tiling honeycomb	hexah	2t{6,3,6}
Rectified order-6 hexagonal tiling honeycomb	rihihexah	r{6,3,6}
Truncated order-6 hexagonal tiling honeycomb	thihexah	t{6,3,6}
Order-6 hexagonal tiling honeycomb	hihexah	{6,3,6}
Small rhombated order-6 hexagonal tiling honeycomb	srihihexah	rr{6,3,6}
Great rhombated order-6 hexagonal tiling honeycomb	grihihexah	tr{6,3,6}
Small rhombated order-6 hexagonal tiling honeycomb	srihihexah	rr{6,3,6}
Great rhombated order-6 hexagonal tiling honeycomb	grihihexah	tr{6,3,6}
Small prismated order-6 hexagonal tiling honeycomb	spiddihexah	t0,3{6,3,6}
Prismatorhombated order-6 hexagonal tiling honeycomb	prihihexah	t0,1,3{6,3,6}
Prismatorhombated order-6 hexagonal tiling honeycomb	prihihexah	t0,1,3{6,3,6}
Great prismated order-6 hexagonal tiling honeycomb	gipiddihexah	t0,1,2,3{6,3,6}

External links

• Klitzing, Richard. "hexah".

• Wikipedia Contributors. "Hexagonal tiling honeycomb".
• lllllllllwith10ls. "Category 1: Regulars" (#5).