Snub triapeirogonal tiling

Snub triapeirogonal tiling

Poincaré disk model of the hyperbolic plane
TypeHyperbolic uniform tiling
Vertex figure3.3.3.3.∞
Schläfli symbolsr{∞,3}
Wythoff symbol| ∞ 3 2
Coxeter diagram
Symmetry group[∞,3]+, (∞32)
DualOrder-3-infinite floret pentagonal tiling
PropertiesVertex-transitive Chiral

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

Images

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

The dual tiling:

Related polyhedra and tiling

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.

Dimensional family of snub polyhedra and tilings: 3.3.3.3.n
Symmetry
n32
[n,3]+		 Spherical Euclidean Compact hyperbolic Paracompact
232
[2,3]+
D3		 332
[3,3]+
T		 432
[4,3]+
O		 532
[5,3]+
I		 632
[6,3]+
P6		 732
[7,3]+		 832
[8,3]+...		 ∞32
[∞,3]+
Snub
figure
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.∞
Coxeter
Schläfli
sr{2,3}
sr{3,3}
sr{4,3}
sr{5,3}
sr{6,3}
sr{7,3}
sr{8,3}
sr{∞,3}
Snub
dual
figure
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.∞
Coxeter
Paracompact hyperbolic uniform tilings in [∞,3] family
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.∞

See also

References

    • John H. Conway, Heidi Burgiel, Chaim Goodman-Strass, 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