Octahedral honeycomb

Octahedral honeycomb
Rank	4
Type	Regular, paracompact
Space	Hyperbolic
Notation
Bowers style acronym	Octh
Coxeter diagram	o4o4o3x ()
Schläfli symbol	{3,4,4}
Elements
Cells	NM octahedra
Faces	4NM triangles
Edges	3NM
Vertices	6N
Vertex figure	Square tiling, edge length 1
Measures (edge length 1)
Circumradius	0
Related polytopes
Army	Octh
Regiment	Octh
Dual	Square tiling honeycomb
Abstract & topological properties
Orientable	Yes
Properties
Symmetry	[4,4,3]
Convex	Yes

The order-4 octahedral honeycomb, or just octahedral honeycomb, is a paracompact regular tiling of 3D hyperbolic space. 4 ideal octahedra meet at each edge. All vertices are ideal points at infinity, with infinitely many octahedra meeting at each vertex in a square tiling arrangement.

Representations

The octahedral honeycomb has the following Coxeter diagrams:

• o4o4o3x (full symmetry)
• o4o4o *b3x (octahedra of two types)
• o3x3o4o4*a (octahedra of three types, x4o4x verf)

Related polytopes

o4o4o3o truncations

Name	OBSA	Schläfli symbol	CD diagram	Picture
Square tiling honeycomb	squah	{4,4,3}
Truncated square tiling honeycomb	tisquah	t{4,4,3}
Rectified square tiling honeycomb	risquah	r{4,4,3}
Octahedral-square tiling honeycomb	osquah	2t{4,4,3}
Rectified octahedral honeycomb	rocth	r{3,4,4}
Truncated octahedral honeycomb	tocth	t{3,4,4}
Octahedral honeycomb	octh	{3,4,4}
Small rhombated square tiling honeycomb	srisquah	rr{4,4,3}
Great rhombated square tiling honeycomb	grisquah	tr{4,4,3}
Small rhombated octahedral honeycomb	srocth	rr{3,4,4}
Great rhombated octahedral honeycomb	grocth	tr{3,4,4}
Small prismated square tiling honeycomb	sidposquah	t0,3{4,4,3}
Prismatorhombated octahedral honeycomb	procth	t0,1,3{4,4,3}
Prismatorhombated square tiling honeycomb	prisquah	t0,1,3{3,4,4}
Great prismated square tiling honeycomb	gidposquah	t0,1,2,3{4,4,3}

External links

• Klitzing, Richard. "octh".

• Wikipedia Contributors. "Order-4 octahedral honeycomb".
• lllllllllwith10ls. "Category 1: Regulars" (#14).