Tetraapeirogonal tiling

In geometry, the tetraapeirogonal tiling is a uniform tiling of the hyperbolic plane with a Schläfli symbol of r{∞,4}.

tetraapeirogonal tiling

Poincaré disk model of the hyperbolic plane
TypeHyperbolic uniform tiling
Vertex configuration(4.∞)2
Schläfli symbolr{∞,4} or
rr{∞,∞} or
Wythoff symbol2 | ∞ 4
∞ | ∞ 2
Coxeter diagram
or
Symmetry group[∞,4], (*∞42)
[∞,∞], (*∞∞2)
DualOrder-4-infinite rhombille tiling
PropertiesVertex-transitive edge-transitive

Uniform constructions

There are 3 lower symmetry uniform construction, one with two colors of apeirogons, one with two colors of squares, and one with two colors of each:

Symmetry (*∞42)
[∞,4]		 (*∞33)
[1+,∞,4] = [(∞,4,4)]		 (*∞∞2)
[∞,4,1+] = [∞,∞]		 (*∞2∞2)
[1+,∞,4,1+]
Coxeter = = =
Schläfli r{∞,4} r{4,∞}1⁄2 r{∞,4}1⁄2=rr{∞,∞} r{∞,4}1⁄4
Coloring
Dual

Symmetry

The dual to this tiling represents the fundamental domains of *∞2∞2 symmetry group. The symmetry can be doubled by adding mirrors on either diagonal of the rhombic domains, creating *∞∞2 and *∞44 symmetry.

*n42 symmetry mutations of quasiregular tilings: (4.n)2
Symmetry
*4n2
[n,4]		 Spherical Euclidean Compact hyperbolic Paracompact Noncompact
*342
[3,4]		 *442
[4,4]		 *542
[5,4]		 *642
[6,4]		 *742
[7,4]		 *842
[8,4]...		 *∞42
[∞,4]		  
[ni,4]
Figures
Config. (4.3)2 (4.4)2 (4.5)2 (4.6)2 (4.7)2 (4.8)2 (4.∞)2 (4.ni)2
Paracompact uniform tilings in [∞,4] family
{∞,4} t{∞,4} r{∞,4} 2t{∞,4}=t{4,∞} 2r{∞,4}={4,∞} rr{∞,4} tr{∞,4}
Dual figures
V∞4 V4.∞.∞ V(4.∞)2 V8.8.∞ V4∞ V43.∞ V4.8.∞
Alternations
[1+,∞,4]
(*44∞)		 [∞+,4]
(∞*2)		 [∞,1+,4]
(*2∞2∞)		 [∞,4+]
(4*∞)		 [∞,4,1+]
(*∞∞2)		 [(∞,4,2+)]
(2*2∞)		 [∞,4]+
(∞42)

=
=
h{∞,4} s{∞,4} hr{∞,4} s{4,∞} h{4,∞} hrr{∞,4} s{∞,4}
Alternation duals
V(∞.4)4 V3.(3.∞)2 V(4.∞.4)2 V3.∞.(3.4)2 V∞∞ V∞.44 V3.3.4.3.∞
Paracompact uniform tilings in [∞,∞] family

=
=
=
=
=
=
=
=
=
=
=
=
{∞,∞} 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.∞

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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