Order-7 triangular tiling
Poincaré disk model of thehyperbolic plane
Type	Hyperbolic regular tiling
Vertex configuration	${\displaystyle 3^7}$
Schläfli symbol	${\displaystyle \{3,7\}}$
Wallpaper group	${\displaystyle [7,3]}$,${\displaystyle {(}^{\ast }732)}$
Dual	heptagonal tiling
Properties	vertex-transitive,edge-transitive,face-transitive

Ingeometry, theorder-7 triangular tiling is aregular tiling of thehyperbolic plane with aSchläfli symbol of {3,7}.

The symmetry group of the tiling is the(2,3,7) triangle group, and a fundamental domain for this action is the (2,3,7)Schwarz triangle. This is the smallest hyperbolic Schwarz triangle, and thus, by the proof ofHurwitz's automorphisms theorem, the tiling is the universal tiling that covers allHurwitz surfaces (theRiemann surfaces with maximal symmetry group), giving them a triangulation whose symmetry group equals their automorphism group as Riemann surfaces.

The smallest of these is theKlein quartic, the most symmetric genus 3 surface, together with a tiling by 56 triangles, meeting at 24 vertices, with symmetry group the simple group of order 168, known asPSL(2,7). The resulting surface can in turn be polyhedrallyimmersed into Euclidean 3-space, yielding thesmall cubicuboctahedron.^[1]

The dualorder-3 heptagonal tiling has the same symmetry group, and thus yields heptagonal tilings of Hurwitz surfaces.

The symmetry group of the order-7 triangular tiling has fundamental domain the (2,3,7)Schwarz triangle, which yields this tiling.

Thesmall cubicuboctahedron is a polyhedral immersion of theKlein quartic,[1] which, like allHurwitz surfaces, is a quotient of this tiling.

It is related to two star-tilings by the samevertex arrangement: theorder-7 heptagrammic tiling, {7/2,7}, andheptagrammic-order heptagonal tiling, {7,7/2}.

This tiling is topologically related as a part of sequence of regular polyhedra withSchläfli symbol {3,p}.

*n32 symmetry mutation of regular tilings: {3,n} v t e
Spherical				Euclid.	Compact hyper.		Paraco.	Noncompact hyperbolic
3.3	33	34	35	36	37	38	3∞	312i	39i	36i	33i

This tiling is a part of regular series {n,7}:

Tiles of the form {n,7}
Spherical	Hyperbolic tilings v t e
{2,7}	{3,7}	{4,7}	{5,7}	{6,7}	{7,7}	{8,7}	...	{∞,7}

From aWythoff construction there are eight hyperbolicuniform 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 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

Wikimedia Commons has media related toOrder-7 triangular tiling.

• Order-7 tetrahedral honeycomb
• List of regular polytopes
• List of uniform planar tilings
• Tilings of regular polygons
• Triangular tiling
• Uniform tilings in hyperbolic plane

• 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

• Hyperbolic tilings
• Isogonal tilings
• Isohedral tilings
• Order-7 tilings
• Regular tilings
• Triangular tilings

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