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Triheptagonal tiling

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Triheptagonal tiling
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In geometry, the triheptagonal tiling is a semiregular tiling of the hyperbolic plane, representing a rectified Order-3 heptagonal tiling. There are two triangles and two heptagons alternating on each vertex. It has Schläfli symbol of r{7,3}.

Triheptagonal tiling
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Poincaré disk model of the hyperbolic plane
TypeHyperbolic uniform tiling
Vertex configuration(3.7)2
Schläfli symbolr{7,3} or
Wythoff symbol2 | 7 3
Coxeter diagram or
Symmetry group[7,3], (*732)
DualOrder-7-3 rhombille tiling
PropertiesVertex-transitive edge-transitive

Compare to trihexagonal tiling with vertex configuration 3.6.3.6.

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Images

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Klein disk model of this tiling preserves straight lines, but distorts angles
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The dual tiling is called an Order-7-3 rhombille tiling, made from rhombic faces, alternating 3 and 7 per vertex.

7-3 Rhombille

Quick Facts 7-3 rhombille tiling, Coxeter diagram ...

In geometry, the 7-3 rhombille tiling is a tessellation of identical rhombi on the hyperbolic plane. Sets of three and seven rhombi meet two classes of vertices.

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7-3 rhombile tiling in band model

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Summarize
Perspective

The triheptagonal tiling can be seen in a sequence of quasiregular polyhedrons and tilings:

More information Sym.*n32 [n,3], Spherical ...

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.

More information Symmetry: [7,3], (*732), [7,3]+, (732) ...
More information Symmetry*7n2 [n,7], Hyperbolic... ...

See also

References

  • 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.


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