Snub_triapeirogonal_tiling Knowpia

Snub triapeirogonal tiling
Poincaré disk model of the hyperbolic plane
Type	Hyperbolic uniform tiling
Vertex configuration	3.3.3.3.∞
Schläfli symbol	sr{∞,3} or $s{\begin{Bmatrix}\infty \\3\end{Bmatrix}}$
Wythoff symbol	\| ∞ 3 2
Coxeter diagram	or
Symmetry group	[∞,3]⁺, (∞32)
Dual	Order-3-infinite floret pentagonal tiling
Properties	Vertex-transitive Chiral

In geometry, the snub triapeirogonal tiling is a uniform tiling of the hyperbolic plane with a Schläfli symbol of sr{∞,3}.

Images edit

Drawn in chiral pairs, with edges missing between black triangles:

The dual tiling:

Related polyhedra and tiling edit

This hyperbolic tiling is topologically related as a part of sequence of uniform snub polyhedra with vertex configurations (3.3.3.3.n), and [n,3] Coxeter group symmetry.

n32 symmetry mutations of snub tilings: 3.3.3.3.n v t e
Symmetry n32	Spherical				Euclidean	Compact hyperbolic		Paracomp.
Symmetry n32	232	332	432	532	632	732	832	∞32
Snub figures
Config.	3.3.3.3.2	3.3.3.3.3	3.3.3.3.4	3.3.3.3.5	3.3.3.3.6	3.3.3.3.7	3.3.3.3.8	3.3.3.3.∞ Gyrofigures
Config.	V3.3.3.3.2	V3.3.3.3.3	V3.3.3.3.4	V3.3.3.3.5	V3.3.3.3.6	V3.3.3.3.7	V3.3.3.3.8	V3.3.3.3.∞

Paracompact uniform tilings in [∞,3] family v t e
Symmetry: [∞,3], (*∞32)							[∞,3]⁺ (∞32)	[1⁺,∞,3] (*∞33)		[∞,3⁺] (3*∞)

		=	=	=				= or	= or	=

{∞,3}	t{∞,3}	r{∞,3}	t{3,∞}	{3,∞}	rr{∞,3}	tr{∞,3}	sr{∞,3}	h{∞,3}	h₂{∞,3}	s{3,∞}
Uniform duals


V∞³	V3.∞.∞	V(3.∞)²	V6.6.∞	V3^∞	V4.3.4.∞	V4.6.∞	V3.3.3.3.∞	V(3.∞)³		V3.3.3.3.3.∞

References edit

John H. Conway, Heidi Burgiel, Chaim Goodman-Strauss, The Symmetries of Things 2008, ISBN 978-1-56881-220-5 (Chapter 19, The Hyperbolic Archimedean Tessellations)
"Chapter 10: Regular honeycombs in hyperbolic space". The Beauty of Geometry: Twelve Essays. Dover Publications. 1999. ISBN 0-486-40919-8. LCCN 99035678.

External links edit

Weisstein, Eric W. "Hyperbolic tiling". MathWorld.
Weisstein, Eric W. "Poincaré hyperbolic disk". MathWorld.

Summary

Images edit

Related polyhedra and tiling edit

See also edit

References edit

External links edit