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Order-6 pentagonal tiling

In this article, we will explore the impact of Order-6 pentagonal tiling on contemporary society. Order-6 pentagonal tiling has been the subject of numerous studies and discussions, generating conflicting opinions and passionate debates. Since its inception, Order-6 pentagonal tiling has captured the attention of researchers, academics and professionals from various areas, becoming a topic of universal interest. In order to fully understand its influence, we will examine its origins, evolution and repercussions on different aspects of daily life. Likewise, we will analyze society's perceptions and attitudes towards Order-6 pentagonal tiling, as well as its impact in the cultural, economic and political sphere. Through this exhaustive analysis, we aim to shed light on a topic that continues to be the subject of analysis and reflection today.

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Order-6 pentagonal tiling
Order-6 pentagonal tiling
Poincaré disk model of the hyperbolic plane
Type Hyperbolic regular tiling
Vertex configuration 56
Schläfli symbol {5,6}
Wythoff symbol 6 | 5 2
Coxeter diagram
Symmetry group , (*652)
Dual Order-5 hexagonal tiling
Properties Vertex-transitive, edge-transitive, face-transitive

In geometry, the order-6 pentagonal tiling is a regular tiling of the hyperbolic plane. It has Schläfli symbol of {5,6}.

Uniform coloring

This regular tiling can also be constructed from symmetry alternating two colors of pentagons, represented by t1(5,5,3).

Symmetry

This tiling represents a hyperbolic kaleidoscope of 6 mirrors defining a regular hexagon fundamental domain, and 5 mirrors meeting at a point. This symmetry by orbifold notation is called *33333 with 5 order-3 mirror intersections.

This tiling is topologically related as a part of sequence of regular tilings with order-6 vertices with Schläfli symbol {n,6}, and Coxeter diagram , progressing to infinity.

Regular tilings {n,6}
Spherical Euclidean Hyperbolic tilings

{2,6}
{3,6}
{4,6}
{5,6}
{6,6}
{7,6}
{8,6}
 ...
{∞,6}
Uniform hexagonal/pentagonal tilings
Symmetry: , (*652) +, (652) , (5*3) , (*553)
{6,5} t{6,5} r{6,5} 2t{6,5}=t{5,6} 2r{6,5}={5,6} rr{6,5} tr{6,5} sr{6,5} s{5,6} h{6,5}
Uniform duals
V65 V5.12.12 V5.6.5.6 V6.10.10 V56 V4.5.4.6 V4.10.12 V3.3.5.3.6 V3.3.3.5.3.5 V(3.5)5

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.

See also

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