Order-∞ apeirogonal tiling

Order-∞ apeirogonal tiling
Rank	3
Type	Regular, paracompact
Space	Hyperbolic
Notation
Bowers style acronym	Azazat
Coxeter diagram	x∞o∞o ()
Schläfli symbol	{∞,∞}
Elements
Faces	2N apeirogons
Edges	NM
Vertices	2N
Vertex figure	Apeirogon, edge length 2
Measures (edge length 1)
Circumradius
Related polytopes
Army	Azazat
Regiment	Azazat
Dual	Order-∞ apeirogonal tiling
Topological properties
Surface	Sphere
Orientable	Yes
Genus	0
Properties
Symmetry	[∞,∞]
Convex	Yes

The order-∞ apeirogonal tiling or infinite-order apeirogonal tiling is a paracompact regular tiling of the hyperbolic plane. Infinitely many apeirogons join at each vertex. It is self-dual.

Representations[edit | edit source]

The order-∞ apeirogonal tiling has the following Coxeter diagrams:

x∞o∞o (full symmetry)
x∞o∞o∞*a (apeirogons of two types) ()

Related polytopes[edit | edit source]

o∞o∞o truncations
Name	OBSA	Schläfli symbol
Order-∞ apeirogonal tiling	azazat	{∞,∞}
Truncated order-∞ apeirogonal tiling = Apeirogonal tiling	azat	t{∞,∞}
Apeiroapeirogonal tiling = Order-4 apeirogonal tiling	squazat	r{∞,∞}
Truncated order-∞ apeirogonal tiling = Apeirogonal tiling	azat	t{∞,∞}
Order-∞ apeirogonal tiling	azazat	{∞,∞}
Small rhombiapeiroapeirogonal tiling = Tetraapeirogonal tiling	tezt	rr{∞,∞}
Great rhombiapeiroapeirogonal tiling = Truncated order-4 apeirogonal tiling	tosquazat	tr{∞,∞}
Snub apeiroapeirogonal tiling		sr{∞,∞}