Snub apeiroapeirogonal tiling

In geometry, the snub apeiroapeirogonal tiling is a uniform tiling of the hyperbolic plane. It has Schläfli symbol of s{∞,∞}. It has 3 equilateral triangles and 2 apeirogons around every vertex, with vertex figure 3.3.∞.3.∞.

Snub apeiroapeirogonal tiling

Poincaré disk model of the hyperbolic plane
TypeHyperbolic uniform tiling
Vertex configuration3.3.∞.3.∞
Schläfli symbols{∞,4}
sr{∞,∞} or
Wythoff symbol| ∞ ∞ 2
Coxeter diagram
or
Symmetry group[∞,∞]+, (∞∞2)
DualInfinitely-infinite-order floret pentagonal tiling
PropertiesVertex-transitive Chiral

Dual tiling

Paracompact uniform tilings in [∞,∞] family

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{∞,∞} t{∞,∞} r{∞,∞} 2t{∞,∞}=t{∞,∞} 2r{∞,∞}={∞,∞} rr{∞,∞} tr{∞,∞}
Dual tilings
V∞∞ V∞.∞.∞ V(∞.∞)2 V∞.∞.∞ V∞∞ V4.∞.4.∞ V4.4.∞
Alternations
[1+,∞,∞]
(*∞∞2)		 [∞+,∞]
(∞*∞)		 [∞,1+,∞]
(*∞∞∞∞)		 [∞,∞+]
(∞*∞)		 [∞,∞,1+]
(*∞∞2)		 [(∞,∞,2+)]
(2*∞∞)		 [∞,∞]+
(2∞∞)
h{∞,∞} s{∞,∞} hr{∞,∞} s{∞,∞} h2{∞,∞} hrr{∞,∞} sr{∞,∞}
Alternation duals
V(∞.∞)∞ V(3.∞)3 V(∞.4)4 V(3.∞)3 V∞∞ V(4.∞.4)2 V3.3.∞.3.∞

The snub tetrapeirogonal tiling is last in an infinite series of snub polyhedra and tilings with vertex figure 3.3.n.3.n.

4n2 symmetry mutations of snub tilings: 3.3.n.3.n
Symmetry
4n2		 Spherical Euclidean Compact hyperbolic Paracompact
222 322 442 552 662 772 882 ∞∞2
Snub
figures
Config. 3.3.2.3.2 3.3.3.3.3 3.3.4.3.4 3.3.5.3.5 3.3.6.3.6 3.3.7.3.7 3.3.8.3.8 3.3.∞.3.∞
Gyro
figures
Config. V3.3.2.3.2 V3.3.3.3.3 V3.3.4.3.4 V3.3.5.3.5 V3.3.6.3.6 V3.3.7.3.7 V3.3.8.3.8 V3.3.∞.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.
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