Truncated tetrapentagonal tiling

Truncated tetrapentagonal tiling
Poincaré disk model of the hyperbolic plane
Type	Hyperbolic uniform tiling
Vertex configuration	4.8.10
Schläfli symbol	tr{5,4} or ${\displaystyle t{\begin{Bmatrix}5\\4\end{Bmatrix}}}$
Wythoff symbol	2 5 4 |
Coxeter diagram	or
Symmetry group	[5,4], (*542)
Dual	Order-4-5 kisrhombille tiling
Properties	Vertex-transitive

In geometry, the truncated tetrapentagonal tiling is a uniform tiling of the hyperbolic plane. It has Schläfli symbol of t_0,1,2{4,5} or tr{4,5}.

There are four small index subgroup constructed from [5,4] by mirror removal and alternation. In these images fundamental domains are alternately colored black and white, and mirrors exist on the boundaries between colors.

A radical subgroup is constructed [5*,4], index 10, as [5⁺,4], (5*2) with gyration points removed, becoming orbifold (*22222), and its direct subgroup [5*,4]⁺, index 20, becomes orbifold (22222).

Small index subgroups of [5,4]
Index	1	2		10
Diagram
Coxeter (orbifold)	[5,4] = (*542)	[5,4,1+] = = (*552)	[5+,4] = (5*2)	[5*,4] = (*22222)
Direct subgroups
Index	2	4		20
Diagram
Coxeter (orbifold)	[5,4]+ = (542)	[5+,4]+ = = (552)		[5*,4]+ = (22222)

*n42 symmetry mutation of omnitruncated tilings: 4.8.2n v t e
Symmetry *n42 [n,4]	Spherical		Euclidean	Compact hyperbolic				Paracomp.
	*242 [2,4]	*342 [3,4]	*442 [4,4]	*542 [5,4]	*642 [6,4]	*742 [7,4]	*842 [8,4]...	*∞42 [∞,4]
Omnitruncated figure	4.8.4	4.8.6	4.8.8	4.8.10	4.8.12	4.8.14	4.8.16	4.8.∞
Omnitruncated duals	V4.8.4	V4.8.6	V4.8.8	V4.8.10	V4.8.12	V4.8.14	V4.8.16	V4.8.∞

*nn2 symmetry mutations of omnitruncated tilings: 4.2n.2n v t e
Symmetry *nn2 [n,n]	Spherical		Euclidean	Compact hyperbolic				Paracomp.
	*222 [2,2]	*332 [3,3]	*442 [4,4]	*552 [5,5]	*662 [6,6]	*772 [7,7]	*882 [8,8]...	*∞∞2 [∞,∞]
Figure
Config.	4.4.4	4.6.6	4.8.8	4.10.10	4.12.12	4.14.14	4.16.16	4.∞.∞
Dual
Config.	V4.4.4	V4.6.6	V4.8.8	V4.10.10	V4.12.12	V4.14.14	V4.16.16	V4.∞.∞

Uniform pentagonal/square tilings v t e
Symmetry: [5,4], (*542)							[5,4]+, (542)	[5+,4], (5*2)	[5,4,1+], (*552)
{5,4}	t{5,4}	r{5,4}	2t{5,4}=t{4,5}	2r{5,4}={4,5}	rr{5,4}	tr{5,4}	sr{5,4}	s{5,4}	h{4,5}
Uniform duals
V54	V4.10.10	V4.5.4.5	V5.8.8	V45	V4.4.5.4	V4.8.10	V3.3.4.3.5	V3.3.5.3.5	V55

Wikimedia Commons has media related to Uniform tiling 4-8-10.

• Uniform tilings in hyperbolic plane
• List of regular polytopes

• 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)
• Coxeter, H. S. M. (1999). "Chapter 10: Regular honeycombs in hyperbolic space". The Beauty of Geometry: Twelve Essays. Dover Publications. ISBN 0-486-40919-8. LCCN 99035678.

• 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

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