11/25/2023 0 Comments Heptagonajd triangle tessellation![]() ![]() ![]() The angles of my hexagon are 120 and three. I have an equilateral triangle with angles of 60 degrees, which is a divisor of four times six. When looking at a regular triangle tessellation, six triangles meet at one vertex. The small cubicuboctahedron is a polyhedral immersion of the Klein quartic, which, like all Hurwitz surfaces, is a quotient of this tiling. These air sketches are just sketches of regular polygons. The name of the tessellation coincides with the number of sides a polygon has that meets with the vertex. ![]() The symmetry group of the order-7 triangular tiling has fundamental domain the (2,3,7) Schwarz triangle, which yields this tiling. The dual order-3 heptagonal tiling has the same symmetry group, and thus yields heptagonal tilings of Hurwitz surfaces. The resulting surface can in turn be polyhedrally immersed into Euclidean 3-space, yielding the small cubicuboctahedron. 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). 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 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. 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. ![]()
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