Truncated triheptagonal tiling

In geometry, the truncated triheptagonal tiling is a semiregular tiling of the hyperbolic plane. There are one square, one hexagon, and one tetradecagon (14-sides) on each vertex. It has Schläfli symbol of tr{7,3}.

Truncated triheptagonal tiling

Poincaré disk model of the hyperbolic plane
TypeHyperbolic uniform tiling
Vertex configuration4.6.14
Schläfli symboltr{7,3} or
Wythoff symbol
Coxeter diagram or
Symmetry group[7,3], (*732)
DualOrder 3-7 kisrhombille
PropertiesVertex-transitive

Uniform colorings

There is only one uniform coloring of a truncated triheptagonal tiling. (Naming the colors by indices around a vertex: 123.)

Symmetry

Each triangle in this dual tiling, order 3-7 kisrhombille, represent a fundamental domain of the Wythoff construction for the symmetry group [7,3].

The dual tiling is called an order-3 bisected heptagonal tiling, made as a complete bisection of the heptagonal tiling, here shown with triangles with alternating colors.

This tiling can be considered a member of a sequence of uniform patterns with vertex figure (4.6.2p) and Coxeter-Dynkin diagram . For p < 6, the members of the sequence are omnitruncated polyhedra (zonohedrons), shown below as spherical tilings. For p > 6, they are tilings of the hyperbolic plane, starting with the truncated triheptagonal tiling.

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.

See also

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.


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