Rhombitriheptagonal tiling

In this article, the impact of Rhombitriheptagonal tiling on contemporary society will be explored. Since its emergence, Rhombitriheptagonal tiling has aroused continuous interest and debate in various fields, whether in politics, popular culture, science or in people's daily lives. Over the years, Rhombitriheptagonal tiling has been the subject of studies and research that have attempted to understand its influence on the way we relate, think, and operate in the world. Without a doubt, Rhombitriheptagonal tiling has been the protagonist of important transformations, generating controversies, admiration and reflections that position it as a relevant topic today. In this sense, it will be interesting to analyze the different aspects surrounding Rhombitriheptagonal tiling and its impact on contemporary society.

Rhombitriheptagonal tiling
Rhombitriheptagonal tiling
Poincaré disk model of the hyperbolic plane
Type Hyperbolic uniform tiling
Vertex configuration 3.4.7.4
Schläfli symbol rr{7,3} or
Wythoff symbol 3 | 7 2
Coxeter diagram or
Symmetry group , (*732)
Dual Deltoidal triheptagonal tiling
Properties Vertex-transitive

In geometry, the rhombitriheptagonal tiling is a semiregular tiling of the hyperbolic plane. At each vertex of the tiling there is one triangle and one heptagon, alternating between two squares. The tiling has Schläfli symbol rr{7, 3}. It can be seen as constructed as a rectified triheptagonal tiling, r{7,3}, as well as an expanded heptagonal tiling or expanded order-7 triangular tiling.

Dual tiling

The dual tiling is called a deltoidal triheptagonal tiling, and consists of congruent kites. It is formed by overlaying an order-3 heptagonal tiling and an order-7 triangular 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.

Uniform heptagonal/triangular tilings
Symmetry: , (*732) +, (732)
{7,3} t{7,3} r{7,3} t{3,7} {3,7} rr{7,3} tr{7,3} sr{7,3}
Uniform duals
V73 V3.14.14 V3.7.3.7 V6.6.7 V37 V3.4.7.4 V4.6.14 V3.3.3.3.7

Symmetry mutations

This tiling is topologically related as a part of sequence of cantellated polyhedra with vertex figure (3.4.n.4), and continues as tilings of the hyperbolic plane. These vertex-transitive figures have (*n32) reflectional symmetry.

*n32 symmetry mutation of dual expanded tilings: V3.4.n.4
Symmetry
*n32
 Spherical Euclid. Compact hyperb. Paraco.
*232
 *332
 *432
 *532
 *632
 *732
 *832
... 		*∞32
Figure
Config.
V3.4.2.4
V3.4.3.4
V3.4.4.4
V3.4.5.4
V3.4.6.4
V3.4.7.4
V3.4.8.4
V3.4.∞.4

See also

Wikimedia Commons has media related to Uniform tiling 3-4-7-4.

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.


Stub icon

This hyperbolic geometry-related article is a stub. You can help Wikipedia by expanding it.