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 CDel node 1.pngCDel 7.pngCDel node 1.pngCDel 4.pngCDel node.png
CDel node 1.pngCDel 7.pngCDel node 1.pngCDel 7.pngCDel node 1.png or CDel node 1.pngCDel split1-77.pngCDel nodes 11.png
Symmetry group [7,4], (*742)
[7,7], (*772)
Dual
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[]

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 heptahexagonal
Image Uniform tiling 74-t01.png Uniform tiling 77-t012.png
Symmetry [7,4]
(*742)
CDel node c1.pngCDel 7.pngCDel node c2.pngCDel 4.pngCDel node c3.png 		[7,7] = [7,4,1+]
(*772)
CDel node c1.pngCDel split1-77.pngCDel nodeab c2.png = CDel node c1.pngCDel 7.pngCDel node c2.pngCDel 4.pngCDel node h0.png
Symbol t{7,4} tr{7,7}
Coxeter diagram CDel node 1.pngCDel 7.pngCDel node 1.pngCDel 4.pngCDel node.png CDel node 1.pngCDel 7.pngCDel node 1.pngCDel 7.pngCDel node 1.png

Symmetry[]

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 772 symmetry 000.png 772 symmetry aaa.png
Coxeter
(orbifold) 		[7,7] = CDel node c1.pngCDel 7.pngCDel node c1.pngCDel 7.pngCDel node c1.png
(*772) 		[7,7]+ = CDel node h2.pngCDel 7.pngCDel node h2.pngCDel 7.pngCDel node h2.png
(772)

Related polyhedra and tiling[]

*n42 symmetry mutation of truncated tilings: 4.2n.2n
  • v
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 		Spherical square prism.png Uniform tiling 432-t12.png Uniform tiling 44-t01.png H2-5-4-trunc-dual.svg H2 tiling 246-3.png H2 tiling 247-3.png H2 tiling 248-3.png H2 tiling 24i-3.png
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 		Spherical square bipyramid.png Spherical tetrakis hexahedron.png 1-uniform 2 dual.svg H2-5-4-kis-primal.svg Order-6 tetrakis square tiling.png Hyperbolic domains 772.png Order-8 tetrakis square tiling.png H2checkers 2ii.png
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
  • v
Symmetry: [7,4], (*742) [7,4]+, (742) [7+,4], (7*2) [7,4,1+], (*772)
CDel node 1.pngCDel 7.pngCDel node.pngCDel 4.pngCDel node.png CDel node 1.pngCDel 7.pngCDel node 1.pngCDel 4.pngCDel node.png CDel node.pngCDel 7.pngCDel node 1.pngCDel 4.pngCDel node.png CDel node.pngCDel 7.pngCDel node 1.pngCDel 4.pngCDel node 1.png CDel node.pngCDel 7.pngCDel node.pngCDel 4.pngCDel node 1.png CDel node 1.pngCDel 7.pngCDel node.pngCDel 4.pngCDel node 1.png CDel node 1.pngCDel 7.pngCDel node 1.pngCDel 4.pngCDel node 1.png CDel node h.pngCDel 7.pngCDel node h.pngCDel 4.pngCDel node h.png CDel node h.pngCDel 7.pngCDel node h.pngCDel 4.pngCDel node.png CDel node.pngCDel 7.pngCDel node.pngCDel 4.pngCDel node h.png
Uniform tiling 74-t0.png Uniform tiling 74-t01.png Uniform tiling 74-t1.png Uniform tiling 74-t12.png Uniform tiling 74-t2.png Uniform tiling 74-t02.png Uniform tiling 74-t012.png Uniform tiling 74-snub.png Uniform tiling 74-h01.png Uniform tiling 77-t0.png
{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
CDel node f1.pngCDel 7.pngCDel node.pngCDel 4.pngCDel node.png CDel node f1.pngCDel 7.pngCDel node f1.pngCDel 4.pngCDel node.png CDel node.pngCDel 7.pngCDel node f1.pngCDel 4.pngCDel node.png CDel node.pngCDel 7.pngCDel node f1.pngCDel 4.pngCDel node f1.png CDel node.pngCDel 7.pngCDel node.pngCDel 4.pngCDel node f1.png CDel node f1.pngCDel 7.pngCDel node.pngCDel 4.pngCDel node f1.png CDel node f1.pngCDel 7.pngCDel node f1.pngCDel 4.pngCDel node f1.png CDel node fh.pngCDel 7.pngCDel node fh.pngCDel 4.pngCDel node fh.png CDel node fh.pngCDel 7.pngCDel node fh.pngCDel 4.pngCDel node.png CDel node.pngCDel 7.pngCDel node.pngCDel 4.pngCDel node fh.png
Uniform tiling 74-t2.png Hyperbolic domains 772.png Ord74 qreg rhombic til.png Order4 heptakis heptagonal til.png Uniform tiling 74-t0.png Deltoidal tetraheptagonal til.png Hyperbolic domains 742.png Uniform tiling 77-t2.png
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
  • v
Symmetry: [7,7], (*772) [7,7]+, (772)
CDel node 1.pngCDel 7.pngCDel node.pngCDel 7.pngCDel node.png = CDel nodes 10ru.pngCDel split2-77.pngCDel node.png
= CDel node h1.pngCDel 4.pngCDel node.pngCDel 7.pngCDel node.png 		CDel node 1.pngCDel 7.pngCDel node 1.pngCDel 7.pngCDel node.png = CDel nodes 10ru.pngCDel split2-77.pngCDel node 1.png
= CDel node h1.pngCDel 4.pngCDel node.pngCDel 7.pngCDel node 1.png 		CDel node.pngCDel 7.pngCDel node 1.pngCDel 7.pngCDel node.png = CDel nodes.pngCDel split2-77.pngCDel node 1.png
= CDel node h0.pngCDel 4.pngCDel node.pngCDel 7.pngCDel node 1.png 		CDel node.pngCDel 7.pngCDel node 1.pngCDel 7.pngCDel node 1.png = CDel nodes 01rd.pngCDel split2-77.pngCDel node 1.png
= CDel node h1.pngCDel 4.pngCDel node.pngCDel 7.pngCDel node 1.png 		CDel node.pngCDel 7.pngCDel node.pngCDel 7.pngCDel node 1.png = CDel nodes 01rd.pngCDel split2-77.pngCDel node.png
= CDel node h1.pngCDel 4.pngCDel node.pngCDel 7.pngCDel node.png 		CDel node 1.pngCDel 7.pngCDel node.pngCDel 7.pngCDel node 1.png = CDel nodes 11.pngCDel split2-77.pngCDel node.png
= CDel node h0.pngCDel 4.pngCDel node 1.pngCDel 7.pngCDel node.png 		CDel node 1.pngCDel 7.pngCDel node 1.pngCDel 7.pngCDel node 1.png =CDel nodes 11.pngCDel split2-77.pngCDel node 1.png
= CDel node h0.pngCDel 4.pngCDel node 1.pngCDel 7.pngCDel node 1.png 		CDel node h.pngCDel 7.pngCDel node h.pngCDel 7.pngCDel node h.png =CDel nodes hh.pngCDel split2-77.pngCDel node h.png
= CDel node h0.pngCDel 4.pngCDel node h.pngCDel 7.pngCDel node h.png
Uniform tiling 77-t0.png Uniform tiling 77-t01.png Uniform tiling 77-t1.png Uniform tiling 77-t12.png Uniform tiling 77-t2.png Uniform tiling 77-t02.png Uniform tiling 77-t012.png Uniform tiling 77-snub.png
{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
CDel node f1.pngCDel 7.pngCDel node.pngCDel 7.pngCDel node.png CDel node f1.pngCDel 7.pngCDel node f1.pngCDel 7.pngCDel node.png CDel node.pngCDel 7.pngCDel node f1.pngCDel 7.pngCDel node.png CDel node.pngCDel 7.pngCDel node f1.pngCDel 7.pngCDel node f1.png CDel node.pngCDel 7.pngCDel node.pngCDel 7.pngCDel node f1.png CDel node f1.pngCDel 7.pngCDel node.pngCDel 7.pngCDel node f1.png CDel node f1.pngCDel 7.pngCDel node f1.pngCDel 7.pngCDel node f1.png CDel node fh.pngCDel 7.pngCDel node fh.pngCDel 7.pngCDel node fh.png
Uniform tiling 77-t2.png Order7 heptakis heptagonal til.png Uniform tiling 74-t2.png Order7 heptakis heptagonal til.png Uniform tiling 77-t0.png Ord74 qreg rhombic til.png Hyperbolic domains 772.png
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[]

  • 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[]

External links[]

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