Pentahexagonal tiling

In geometry, the pentahexagonal tiling is a uniform tiling of the hyperbolic plane. It has Schläfli symbol of r{6,5} or t₁{6,5}.

Pentahexagonal tiling
Poincaré disk model of the hyperbolic plane
Type	Hyperbolic uniform tiling
Vertex configuration	(5.6^2
Schläfli symbol	r{6,5} or ${\displaystyle {\begin{Bmatrix}6\\5\end{Bmatrix}}}$
Wythoff symbol	6 5
Coxeter diagram
Symmetry group	[6,5], (*652)
Dual	Order-6-5 rhombille tiling
Properties	Vertex-transitive edge-transitive

Uniform colorings

Related polyhedra and tiling

Uniform hexagonal/pentagonal tilings
Symmetry: [6,5], (*652)							[6,5]^+, (652)	[6,5^+], (5*3)	[1^+,6,5], (*553)
{6,5}	t{6,5}	r{6,5}	2t{6,5}=t{5,6}	2r{6,5}={5,6}	rr{6,5}	tr{6,5}	sr{6,5}	s{5,6}	h{6,5}
Uniform duals
V6^5	V5.12.12	V5.6.5.6	V6.10.10	V5^6	V4.5.4.6	V4.10.12	V3.3.5.3.6	V3.3.3.5.3.5	V(3.5)^5

*5n2 symmetry mutations of quasiregular tilings: (5.n)^2
Symmetry *5n2 [n,5]	Spherical	Hyperbolic					Paracompact	Noncompact
	*352 [3,5]	*452 [4,5]	*552 [5,5]	*652 [6,5]	*752 [7,5]	*852 [8,5]...	*∞52 [∞,5]	[ni,5]
Figures
Config.	(5.3)^2	(5.4)^2	(5.5)^2	(5.6)^2	(5.7)^2	(5.8)^2	(5.∞)^2	(5.ni)^2
Rhombic figures
Config.	V(5.3)^2	V(5.4)^2	V(5.5)^2	V(5.6)^2	V(5.7)^2	V(5.8)^2	V(5.∞)^2	V(5.∞)^2

Symmetry mutation of quasiregular tilings: 6.n.6.n
Symmetry *6n2 [n,6]	Euclidean	Compact hyperbolic					Paracompact	Noncompact
	*632 [3,6]	*642 [4,6]	*652 [5,6]	*662 [6,6]	*762 [7,6]	*862 [8,6]...	*∞62 [∞,6]	[iπ/λ,6]
Quasiregular figures configuration	6.3.6.3	6.4.6.4	6.5.6.5	6.6.6.6	6.7.6.7	6.8.6.8	6.∞.6.∞	6.∞.6.∞
Dual figures
Rhombic figures configuration	V6.3.6.3	V6.4.6.4	V6.5.6.5	V6.6.6.6	V6.7.6.7	V6.8.6.8	V6.∞.6.∞

[(5,5,3)] reflective symmetry uniform tilings

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.

See also

• Square tiling
• Tilings of regular polygons
• List of uniform planar tilings
• List of regular polytopes

External links

• Weisstein, Eric W. "Hyperbolic tiling". MathWorld.
• Weisstein, Eric W. "Poincaré hyperbolic disk". MathWorld.
• Hyperbolic and Spherical Tiling Gallery
• KaleidoTile 3: Educational software to create spherical, planar and hyperbolic tilings
• Hyperbolic Planar Tessellations, Don Hatch