Order-7 heptagrammic tiling

Order-7 heptagrammic tiling
Poincaré disk model of the hyperbolic plane
Type	Hyperbolic regular tiling
Vertex figure	(7/2)7
Schläfli symbol	{7/2,7}
Wythoff symbol	7 | 7/2 2
Coxeter diagram
Symmetry group	[7,3], (*732)
Dual	Heptagrammic-order heptagonal tiling
Properties	Vertex-transitive, edge-transitive, face-transitive

In geometry, the order-7 heptagrammic tiling is a tiling of the hyperbolic plane by overlapping heptagrams.

Description

This tiling is a regular star-tiling, and has Schläfli symbol of {7/2,7}. The heptagrams forming the tiling are of type {7/2}, . The overlapping heptagrams subdivide the hyperbolic plane into isosceles triangles, 14 of which form each heptagram.

Each point of the hyperbolic plane that does not lie on a heptagram edge belongs to the central heptagon of one heptagram, and is in one of the points of exactly one other heptagram. The winding number of each heptagram around its points is one, and the winding number around the central heptagon is two, so adding these two numbers together, each point of the plane is surrounded three times; that is, the density of the tiling is 3.

In the Euclidean plane, a heptagram of type {7/2} would have angles of 3π/7 at its vertices, but in the hyperbolic plane heptagrams can have the sharper vertex angle 2π/7 that is needed to make exactly seven other heptagrams meet up at the center of each heptagram of the tiling.

Related tilings

It has the same vertex arrangement as the regular order-7 triangular tiling, {3,7}. The full set of edges coincide with the edges of a heptakis heptagonal tiling. The valance 6 vertices in this tiling are false-vertices in the heptagrammic one caused by crossed edges.

It is related to a Kepler-Poinsot polyhedron, the small stellated dodecahedron, {5/2,5}, which is polyhedron and a density-3 regular star-tiling on the sphere:

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

Wikimedia Commons has media related to Order-7 heptagrammic tiling.

External links

• Weisstein, Eric W. "Hyperbolic tiling". MathWorld. 

• Weisstein, Eric W. "Poincaré hyperbolic disk". MathWorld. 

Tessellation
Periodic Pythagorean Rhombille Schwarz triangle Rectangle Domino Uniform tiling & honeycomb Coloring Convex Kisrhombille Wallpaper group Wythoff
Aperiodic Ammann–Beenker Aperiodic set of prototiles List Einstein problem Socolar–Taylor Gilbert Penrose Pentagonal Pinwheel Quaquaversal Rep-tile & Self-tiling Sphinx Truchet
Other Anisohedral & Isohedral Architectonic & catoptric Circle Limit III Computer graphics Honeycomb Isotoxal Packing Problems Domino Wang Heesch's Squaring Prototile Conway criterion Girih Regular Division of the Plane Regular grid Substitution Voronoi Voderberg
By vertex type Spherical 2n 33.n V33.n 42.n V42.n Regular 2∞ 36 44 63 Semi- regular 32.4.3.4 V32.4.3.4 33.42 33.∞ 34.6 V34.6 3.4.6.4 (3.6)2 3.122 42.∞ 4.6.12 4.82 Hyper- bolic 32.4.3.5 32.4.3.6 32.4.3.7 32.4.3.8 32.4.3.∞ 32.5.3.5 32.5.3.6 32.6.3.6 32.6.3.8 32.7.3.7 32.8.3.8 33.4.3.4 32.∞.3.∞ 34.7 34.8 34.∞ 35.4 37 38 3∞ (3.4)3 (3.4)4 3.4.62.4 3.4.7.4 3.4.8.4 3.4.∞.4 3.6.4.6 (3.7)2 (3.8)2 3.142 3.162 (3.∞)2 3.∞2 42.5.4 42.6.4 42.7.4 42.8.4 42.∞.4 45 46 47 48 4∞ (4.5)2 (4.6)2 4.6.12 4.6.14 V4.6.14 4.6.16 V4.6.16 4.6.∞ (4.7)2 (4.8)2 4.8.10 V4.8.10 4.8.12 4.8.14 4.8.16 4.8.∞ 4.102 4.10.12 4.122 4.12.16 4.142 4.162 4.∞2 (4.∞)2 54 55 56 5∞ 5.4.6.4 (5.6)2 5.82 5.102 5.122 (5.∞)2 64 65 66 68 6.4.8.4 (6.8)2 6.82 6.102 6.122 6.162 73 74 77 7.62 7.82 7.142 83 84 86 88 812 8.62 8.162 ∞3 ∞4 ∞5 ∞∞ ∞.62 ∞.82