Triheptagonal tiling

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
Triheptagonal tiling
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.

Images

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

7-3 Rhombille
7-3 rhombille tiling
FacesRhombi
Coxeter diagram
Symmetry group[7,3], *732
Rotation group[7,3]+, (732)
Dual polyhedronTriheptagonal tiling
Face configurationV3.7.3.7
Propertiesedge-transitive face-transitive

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.


7-3 rhombile tiling in band model

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

Quasiregular tilings: (3.n)2
Sym.
*n32
[n,3]		 Spherical Euclid. Compact hyperb. Paraco. Noncompact hyperbolic
*332
[3,3]
Td		 *432
[4,3]
Oh		 *532
[5,3]
Ih		 *632
[6,3]
p6m		 *732
[7,3]
 		 *832
[8,3]...
 		 *∞32
[∞,3]
 		 [12i,3] [9i,3] [6i,3]
Figure
Figure
Vertex (3.3)2 (3.4)2 (3.5)2 (3.6)2 (3.7)2 (3.8)2 (3.∞)2 (3.12i)2 (3.9i)2 (3.6i)2
Schläfli r{3,3} r{3,4} r{3,5} r{3,6} r{3,7} r{3,8} r{3,∞} r{3,12i} r{3,9i} r{3,6i}
Coxeter
Dual uniform figures
Dual
conf.
V(3.3)2
V(3.4)2
V(3.5)2
V(3.6)2
V(3.7)2
V(3.8)2
V(3.∞)2

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: [7,3], (*732) [7,3]+, (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
Dimensional family of quasiregular polyhedra and tilings: 7.n.7.n
Symmetry
*7n2
[n,7]		 Hyperbolic... Paracompact Noncompact
*732
[3,7]		 *742
[4,7]		 *752
[5,7]		 *762
[6,7]		 *772
[7,7]		 *872
[8,7]...		 *∞72
[∞,7]		  
[iπ/λ,7]
Coxeter
Quasiregular
figures
configuration
3.7.3.7
4.7.4.7
7.5.7.5
7.6.7.6
7.7.7.7
7.8.7.8
7.∞.7.∞		  
7.∞.7.∞

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

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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