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Truncated order-4 heptagonal tiling

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Truncated heptagonal tiling
Truncated order-4 heptagonal tiling
Poincaré disk model of the hyperbolic plane
Type Hyperbolic uniform tiling
Vertex configuration 4.14.14
Schläfli symbol t{7,4}
Wythoff symbol 2 4 | 7
2 7 7 |
Coxeter diagram
or
Symmetry group [7,4], (*742)
[7,7], (*772)
Dual Order-7 tetrakis square tiling
Properties Vertex-transitive

In geometry, the truncated order-4 heptagonal tiling is a uniform tiling of the hyperbolic plane. It has Schläfli symbol of t{7,4}.

Constructions

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There are two uniform constructions of this tiling, first by the [7,4] kaleidoscope, and second by removing the last mirror, [7,4,1+], gives [7,7], (*772).

Two uniform constructions of 4.7.4.7
Name Tetraheptagonal Truncated heptaheptagonal
Image
Symmetry [7,4]
(*742)
 [7,7] = [7,4,1+]
(*772)
=
Symbol t{7,4} tr{7,7}
Coxeter diagram

Symmetry

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There is only one simple subgroup [7,7]+, index 2, removing all the mirrors. This symmetry can be doubled to 742 symmetry by adding a bisecting mirror.

Small index subgroups of [7,7]
Type Reflectional Rotational
Index 1 2
Diagram
Coxeter
(orbifold) 		[7,7] =
(*772) 		[7,7]+ =
(772)
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*n42 symmetry mutation of truncated tilings: 4.2n.2n
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]
Truncated
figures
Config. 4.4.4 4.6.6 4.8.8 4.10.10 4.12.12 4.14.14 4.16.16 4.∞.∞
n-kis
figures
Config. V4.4.4 V4.6.6 V4.8.8 V4.10.10 V4.12.12 V4.14.14 V4.16.16 V4.∞.∞
Uniform heptagonal/square tilings
Symmetry: [7,4], (*742) [7,4]+, (742) [7+,4], (7*2) [7,4,1+], (*772)
{7,4} t{7,4} r{7,4} 2t{7,4}=t{4,7} 2r{7,4}={4,7} rr{7,4} tr{7,4} sr{7,4} s{7,4} h{4,7}
Uniform duals
V74 V4.14.14 V4.7.4.7 V7.8.8 V47 V4.4.7.4 V4.8.14 V3.3.4.3.7 V3.3.7.3.7 V77
Uniform heptaheptagonal tilings
Symmetry: [7,7], (*772) [7,7]+, (772)
=
= 		=
= 		=
= 		=
= 		=
= 		=
= 		=
= 		=
=
{7,7} t{7,7}
 r{7,7} 2t{7,7}=t{7,7} 2r{7,7}={7,7} rr{7,7} tr{7,7} sr{7,7}
Uniform duals
V77 V7.14.14 V7.7.7.7 V7.14.14 V77 V4.7.4.7 V4.14.14 V3.3.7.3.7

References

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  • 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.

See also

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Wikimedia Commons has media related to Uniform tiling 4-14-14.
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