Snub triheptagonal tiling

In geometry, the order-3 snub heptagonal tiling is a semiregular tiling of the hyperbolic plane. There are four triangles, one heptagon on each vertex. It has Schläfli symbol of sr{7,3}. The snub tetraheptagonal tiling is another related hyperbolic tiling with Schläfli symbol sr{7,4}.

Snub triheptagonal tiling

Poincaré disk model of the hyperbolic plane
TypeHyperbolic uniform tiling
Vertex configuration3.3.3.3.7
Schläfli symbolsr{7,3} or
Wythoff symbol| 7 3 2
Coxeter diagram or
Symmetry group[7,3]+, (732)
DualOrder-7-3 floret pentagonal tiling
PropertiesVertex-transitive Chiral

Images

Drawn in chiral pairs, with edges missing between black triangles:

Dual tiling

The dual tiling is called an order-7-3 floret pentagonal tiling, and is related to the floret pentagonal tiling.

This semiregular tiling is a member of a sequence of snubbed polyhedra and tilings with vertex figure (3.3.3.3.n) and Coxeter–Dynkin diagram . These figures and their duals have (n32) rotational symmetry, being in the Euclidean plane for n=6, and hyperbolic plane for any higher n. The series can be considered to begin with n=2, with one set of faces degenerated into digons.

From a Wythoff construction there are eight hyperbolic uniform tilings that can be based from the regular heptagonal tiling.

Drawing the tiles colored as red on the original faces, yellow at the original vertices, and blue along the original edges, there are 8 forms.

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
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