 Order7 triangular tiling

Order7 triangular tiling
Poincaré disk model of the hyperbolic planeType Regular hyperbolic tiling Vertex figure 3^{7} Schläfli symbol(s) {3,7} Wythoff symbol(s) 7  3 2 CoxeterDynkin(s) Coxeter group [7,3] Dual Order3 heptagonal tiling Properties Vertextransitive, edgetransitive, facetransitive In geometry, the order7 triangular tiling is a regular tiling of the hyperbolic plane with a Schläfli symbol of {3,7}.
Contents
Related polyhedra and tiling
This tiling is topologically related as a part of sequence of regular polyhedra with Schläfli symbol {3,p}.
{3,3}
{3,4}
{3,5}
{3,6}
{3,7}
{3,8}
{3,9}The dual tiling is the order3 heptagonal tiling.
order3 heptagonal tiling
order7 triangular tilingWythoff constructions yields further uniform tilings, yielding eight uniform tilings, including the two regular ones given here.
Hurwitz surfaces
Further information: Hurwitz surfaceThe 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 of Hurwitz's automorphisms theorem, the tiling is the universal tiling that covers all Hurwitz surfaces (the Riemann surfaces with maximal symmetry group), giving them a triangulation whose symmetry group equals their automorphism group as Riemann surfaces.
The smallest of these is the Klein 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 as PSL(2,7). The resulting surface can in turn be polyhedrally immersed into Euclidean 3space, yielding the small cubicuboctahedron.^{[1]}
The dual order3 heptagonal tiling has the same symmetry group, and thus yields heptagonal tilings of Hurwitz surfaces.
References
 ^ ^{a} ^{b} (Richter) Note each face in the polyhedron consist of multiple faces in the tiling – two triangular faces constitute a square face and so forth, as per this explanatory image.
 John H. Conway, Heidi Burgiel, Chaim GoodmanStrass, The Symmetries of Things 2008, ISBN 9781568812205 (Chapter 19, The Hyperbolic Archimedean Tessellations)
 The Beauty of Geometry: Twelve Essays (1999), Dover Publications, LCCN 9935678, ISBN 0486409198 (Chapter 10: Regular honeycombs in hyperbolic space)
 Richter, David A., How to Make the Mathieu Group M_{24}, http://homepages.wmich.edu/~drichter/mathieu.htm, retrieved 20100415
See also
 List of regular polytopes
 List of uniform planar tilings
 Tilings of regular polygons
 Triangular tiling
 Uniform tilings in hyperbolic plane
External links
 Weisstein, Eric W., "Hyperbolic tiling" from MathWorld.
 Weisstein, Eric W., "Poincaré hyperbolic disk" from 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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