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
Poincaré disk model of the hyperbolic plane
Type	Hyperbolic uniform tiling
Vertex configuration	(3.7)2
Schläfli symbol	r{7,3} or ${\displaystyle {\begin{Bmatrix}7\\3\end{Bmatrix}}}$
Wythoff symbol	2 | 7 3
Coxeter diagram	or
Symmetry group	[7,3], (*732)
Dual	Order-7-3 rhombille tiling
Properties	Vertex-transitive edge-transitive

Compare to trihexagonal tiling with vertex configuration 3.6.3.6.

Quick Facts 7-3 rhombille tiling, Faces ...

7-3 rhombille tiling
Faces	Rhombi
Coxeter diagram
Symmetry group	[7,3], *732
Rotation group	[7,3]+, (732)
Dual polyhedron	Triheptagonal tiling
Face configuration	V3.7.3.7
Properties	edge-transitive face-transitive

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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:

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

Quasiregular tilings: (3.n)2 v t e
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

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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) ...

Uniform heptagonal/triangular tilings v t e
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

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More information Symmetry*7n2 [n,7], Hyperbolic... ...

Dimensional family of quasiregular polyhedra and tilings: 7.n.7.n v t e
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.∞

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

• Trihexagonal tiling - 3.6.3.6 tiling
  • Rhombille tiling - dual V3.6.3.6 tiling
• Tilings of regular polygons
• List of uniform tilings

• 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.

• Weisstein, Eric W. "Hyperbolic tiling". MathWorld.
• Weisstein, Eric W. "Poincaré hyperbolic disk". MathWorld.
• Hyperbolic and Spherical Tiling Gallery
• KaleidoTile 3: Educational software to create spherical, planar and hyperbolic tilings
• Hyperbolic Planar Tessellations, Don Hatch

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