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Coxeter decompositions of hyperbolic polygons information


A hyperbolic triangle – its interior angles do not sum to 180 degrees.

A Coxeter decomposition of a polygon is a decomposition into a finite number of polygons in which any two sharing a side are reflections of each other along that side. Hyperbolic polygons are the analogues of Euclidean polygons in hyperbolic geometry. A hyperbolic n-gon is an area bounded by n segments, rays, or entire straight lines. The standard model for this geometry is the Poincaré disk model. A major difference between Euclidean and hyperbolic polygons is that the sum of internal angles of a hyperbolic polygon is not the same as Euclidean polygons. In particular, the sum of the angles of a hyperbolic triangle is less than 180 degrees.

A triangle group formed by reflecting a triangle on its sides.

Coxeter decompositions are named after Harold Scott MacDonald Coxeter, an accomplished 20th century geometer. He introduced the Coxeter group, an abstract group generated by reflections. These groups have many uses, including producing the rotations of Platonic solids and tessellating the plane.

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Coxeter decompositions of hyperbolic polygons

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A Coxeter decomposition of a polygon is a decomposition into a finite number of polygons in which any two sharing a side are reflections of each other...

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Apeirogon

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results of blending the 1-dimensional apeirogon with other polygons.: 231  Since every polygon is a quotient of the apeirogon, the blend of any polygon with...

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dimensions. There also exist hyperbolic Lorentzian cocompact groups where removing any permutation of two nodes in its Coxeter–Dynkin diagram leaves a finite...

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Polytope

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in non-Euclidean spaces such as hyperbolic space. An important milestone was reached in 1948 with H. S. M. Coxeter's book Regular Polytopes, summarizing...

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Polyhedron

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polyhedra are nonconvex. They are the 3D analogs of 2D orthogonal polygons, also known as rectilinear polygons. Orthogonal polyhedra are used in computational...

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4 21 polytope

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symmetry of the E8 group. It was discovered by Thorold Gosset, published in his 1900 paper. He called it an 8-ic semi-regular figure. Its Coxeter symbol...

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Ideal polyhedron

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three-dimensional hyperbolic space. It can be defined as the convex hull of a finite set of ideal points. An ideal polyhedron has ideal polygons as its faces...

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Schwarz triangle

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tessellation of the Schwarz triangles can be viewed as a generalization of the theory of infinite Coxeter groups, following the theory of hyperbolic reflection...

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Dehn invariant

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from the early 19th century states that any two polygons of equal area can be cut up into polygonal pieces and reassembled into each other. In the late...

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Incidence geometry

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such as projective planes, affine planes, generalized polygons, partial geometries and near polygons. Very general incidence structures can be obtained by...

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Cyclic group

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Example: Subgroups of Cyclic Groups). (Gannon 2006, p. 18). (Gallian 2010, p. 84, Exercise 43). (Jungnickel 1992, pp. 545–547). (Coxeter & Moser 1980, p...

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Algebraic geometry

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written as the union of two smaller algebraic sets. Any algebraic set is a finite union of irreducible algebraic sets and this decomposition is unique. Thus...

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Complex geometry

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This is a polygon in R n {\displaystyle \mathbb {R} ^{n}} with the property that any vertex may be put into the standard form of the vertex of the positive...

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List of algorithms

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elementary functions using a table of logarithms CORDIC: computes hyperbolic and trigonometric functions using a table of arctangents Exponentiation: Addition-chain...

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Orbifold

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noting that the star and link of a vertex i of X ', corresponding to a simplex σ of X, have natural decompositions: the star is isomorphic to the abstract...

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Timeline of mathematics

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introduces the idea of thermodynamic simulated annealing algorithms. 1955 – H. S. M. Coxeter et al. publish the complete list of uniform polyhedron. 1955 –...

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Golden ratio

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JSTOR 2969130. Coxeter, H.S.M.; du Val, Patrick; Flather, H.T.; Petrie, J.F. (1938). The Fifty-Nine Icosahedra. Vol. 6. University of Toronto Studies...

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