Square tiling honeycomb

Square tiling honeycomb
Rank	4
Type	Regular, paracompact
Space	Hyperbolic
Notation
Bowers style acronym	Squah
Coxeter diagram	x4o4o3o ()
Schläfli symbol	{4,4,3}
Elements
Cells	6N square tilings
Faces	3NM squares
Edges	4NM
Vertices	NM
Vertex figure	Cube, edge length √2
Measures (edge length 1)
Circumradius	${\frac {i{\sqrt {2}}}{2}}\approx 0.70711i$
Related polytopes
Army	Squah
Regiment	Squah
Dual	Octahedral honeycomb
Abstract & topological properties
Flag count	48NM
Orientable	Yes
Properties
Symmetry	[4,4,3]
Convex	Yes

The order-3 square tiling honeycomb or just square tiling honeycomb is a paracompact regular tiling of 3D hyperbolic space. Each cell is a square tiling whose vertices lie on a horosphere, a surface in hyperbolic space that approaches a single ideal point at infinity. 3 square tilings meet at each edge, and 6 meet at each vertex.

It can be formed by rectifying the order-4 square tiling honeycomb.

Representations

The square tiling honeycomb has the following Coxeter diagrams:

• x4o4o3o () (full symmetry)
• o4x4o4o () (as rectified order-4 square tiling honeycomb)
• o4x4o2o4*b () (cuboid verf)
• x4o4x2o4*b () (square frustum verf)
• x4o4x4o4*a () (rectangular frustum verf)

Related polytopes

o4o4o4o truncations

Name	OBSA	Schläfli symbol	CD diagram	Picture
Order-4 square tiling honeycomb	sisquah	{4,4,4}
Truncated order-4 square tiling honeycomb	tissish	t{4,4,4}
Rectified order-4 square tiling honeycomb = Square tiling honeycomb	squah	r{4,4,4}
Dissquare tiling honeycomb	dish	2t{4,4,4}
Rectified order-4 square tiling honeycomb = Square tiling honeycomb	squah	r{4,4,4}
Truncated order-4 square tiling honeycomb	tissish	t{4,4,4}
Order-4 square tiling honeycomb	sisquah	{4,4,4}
Small rhombated square tiling honeycomb = Rectified square tiling honeycomb	risquah	rr{4,4,4}
Great rhombated square tiling honeycomb = Truncated square tiling honeycomb	tisquah	tr{4,4,4}
Small rhombated square tiling honeycomb = Rectified square tiling honeycomb	risquah	rr{4,4,4}
Great rhombated square tiling honeycomb = Truncated square tiling honeycomb	tisquah	tr{4,4,4}
Small prismated order-4 square tiling honeycomb	spiddish	t0,3{4,4,4}
Prismatorhombated order-4 square tiling honeycomb	prissish	t0,1,3{4,4,4}
Prismatorhombated order-4 square tiling honeycomb	prissish	t0,1,3{4,4,4}
Great prismated order-4 square tiling honeycomb	dipiddish	t0,1,2,3{4,4,4}

External links

• Klitzing, Richard. "squah".

• Wikipedia contributors. "Square tiling honeycomb".
• lllllllllwith10ls. "Category 1: Regulars" (#13).

• lllllllllwith10ls. "Category 4: Square Rectates" (#13 - recount under sisquah symmetry).