Snub order-6 square tiling

Snub order-6 square tiling
Poincaré disk model of the hyperbolic plane
Type	Hyperbolic uniform tiling
Vertex configuration	3.3.3.4.3.4
Schläfli symbol	s(4,4,3) s{4,6}
Wythoff symbol	\| 4 4 3
Coxeter diagram
Symmetry group	[(4,4,3)]⁺, (443) [6,4⁺], (4*3)
Dual	Order-4-4-3 snub dual tiling
Properties	Vertex-transitive

In geometry, the snub order-6 square tiling is a uniform tiling of the hyperbolic plane. It has Schläfli symbol of s{(4,4,3)} or s{4,6}.

Images

Symmetry

The symmetry is doubled as a snub order-6 square tiling, with only one color of square. It has Schläfli symbol of s{4,6}.

Related polyhedra and tiling

The vertex figure 3.3.3.4.3.4 does not uniquely generate a uniform hyperbolic tiling. Another with quadrilateral fundamental domain (3 2 2 2) and 2*32 symmetry is generated by :

Uniform (4,4,3) tilings v t e
Symmetry: [(4,4,3)] (*443)							[(4,4,3)]⁺ (443)	[(4,4,3⁺)] (3*22)	[(4,1⁺,4,3)] (*3232)



h{6,4} t₀(4,4,3)	h₂{6,4} t_0,1(4,4,3)	{4,6}¹/₂ t₁(4,4,3)	h₂{6,4} t_1,2(4,4,3)	h{6,4} t₂(4,4,3)	r{6,4}¹/₂ t_0,2(4,4,3)	t{4,6}¹/₂ t_0,1,2(4,4,3)	s{4,6}¹/₂ s(4,4,3)	hr{4,6}¹/₂ hr(4,3,4)	h{4,6}¹/₂ h(4,3,4)	q{4,6} h₁(4,3,4)
Uniform duals

V(3.4)⁴	V3.8.4.8	V(4.4)³	V3.8.4.8	V(3.4)⁴	V4.6.4.6	V6.8.8	V3.3.3.4.3.4	V(4.4.3)²	V6⁶	V4.3.4.6.6

Uniform tetrahexagonal tilings v t e
*Symmetry: [6,4], (642)** (with [6,6] (662), [(4,3,3)] (443) , [∞,3,∞] (3222) index 2 subsymmetries) (And [(∞,3,∞,3)] (3232) index 4 subsymmetry)
= = =	=	= = =	=	= = =	=

{6,4}	t{6,4}	r{6,4}	t{4,6}	{4,6}	rr{6,4}	tr{6,4}
Uniform duals


V6⁴	V4.12.12	V(4.6)²	V6.8.8	V4⁶	V4.4.4.6	V4.8.12
Alternations
[1⁺,6,4] (*443)	[6⁺,4] (6*2)	[6,1⁺,4] (*3222)	[6,4⁺] (4*3)	[6,4,1⁺] (*662)	[(6,4,2⁺)] (2*32)	[6,4]⁺ (642)
=	=	=	=	=	=

h{6,4}	s{6,4}	hr{6,4}	s{4,6}	h{4,6}	hrr{6,4}	sr{6,4}

Footnotes

References

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.