In geometry, a Schwarz triangle, named after Hermann Schwarz, is a spherical triangle that can be used to tile a sphere (spherical tiling), possibly overlapping...

the acute and obtuse Schwarz triangles. The numbers that can be used for the sides of a non-dihedral acute or obtuse Schwarz triangle that does not necessarily...

complex analysis, the Schwarz triangle function or Schwarz s-function is a function that conformally maps the upper half plane to a triangle in the upper half...

to the Schwarzian derivative (Hille 1976, pp. 307–401). The Schwarz triangle maps or Schwarz s-functions are ratios of pairs of solutions. s k ( z ) = ϕ...

construct the triangle group, start with a hyperbolic triangle with angles π/2, π/3, and π/7. This triangle, the smallest hyperbolic Schwarz triangle, tiles...

triangle can be an ordinary Euclidean triangle, a triangle on the sphere, or a hyperbolic triangle. Each triangle group is the symmetry group of a tiling...

functions. The upper half-plane is mapped to a triangle with circular arcs for edges by the Schwarz triangle map. The upper half-plane is mapped to the square...

theorem Schwarz reflection principle Schwarz triangle Schwarz triangle function Cauchy–Schwarz inequality Theorem of Pohlke and Schwarz Schwarz, H. A....

exceptional Schwarz triangles (triangles that tile the sphere, or more generally Euclidean plane or hyperbolic plane via their triangle group of reflections...

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...

Wythoff construction of a uniform polyhedron or plane tiling within a Schwarz triangle. It was first used by Coxeter, Longuet-Higgins and Miller in their...

A Reuleaux triangle [ʁœlo] is a curved triangle with constant width, the simplest and best known curve of constant width other than the circle. It is formed...

The Cauchy–Schwarz inequality (also called Cauchy–Bunyakovsky–Schwarz inequality) is an upper bound on the inner product between two vectors in an inner...

trigonometry Great-circle distance or spherical distance Lenart sphere Schwarz triangle Spherical geometry Spherical polyhedron Triangulation (surveying) Todhunter...

diverge. The Schwarz lantern demonstrates that sampling a curved surface by close-together points and connecting them by small triangles is inadequate...

well-defined densities. The nonconvex forms are constructed from Schwarz triangles. All the uniform polyhedra are listed below by their symmetry groups...

In geometry, an isosceles triangle (/aɪˈsɒsəliːz/) is a triangle that has two sides of equal length and two angles of equal measure. Sometimes it is specified...

In mathematics, the triangle inequality states that for any triangle, the sum of the lengths of any two sides must be greater than or equal to the length...

t-notation. Simple tiles are generated by Möbius triangles with whole numbers p,q,r, while Schwarz triangles allow rational numbers p,q,r and allow star polygon...

with the property of tiling space only aperiodically. A Schwarz triangle is a spherical triangle that can be used to tile a sphere. It is possible to tessellate...

contexts, the "(2,3,7) triangle group" most often refers, not to the full triangle group Δ(2,3,7) (the Coxeter group with Schwarz triangle (2,3,7) or a realization...

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...

equivalent to the hypergeometric differential equation and f(z) is the Schwarz triangle function, which can be written in terms of hypergeometric functions...

constructed from Schwarz triangle (3 3 5⁄2) with Wythoff symbol 3 | 5⁄2 3. Its hexagonal vertex figure alternates equilateral triangle and pentagram faces...

triangle Regiomontanus Regiomontanus' angle maximization problem Reuschle's theorem Right triangle Routh's theorem Scalene triangle Schwarz triangle Schiffler's...

of uniform polyhedra by Wythoff symbol List of uniform polyhedra by Schwarz triangle Coxeter, Harold Scott MacDonald; Longuet-Higgins, M. S.; Miller, J...

of uniform polyhedra by Wythoff symbol List of uniform polyhedra by Schwarz triangle List of Johnson solids List of Wenninger polyhedron models Polyhedron...

domain triangle, the symmetry group is a hyperbolic triangle group. There are also infinitely many uniform tilings that cannot be generated from Schwarz triangles...

of geodesic domes. It has four Wythoff constructions between four Schwarz triangle families: 2 | 5 5/2, 2 | 5 5/3, 2 | 5/2 5/4, 2 | 5/3 5/4, but represent...

of 5 items. There are only 44 kinds of Schwarz triangles, aside from the infinite dihedral family of triangles (p 2 2) with p = {2, 3, 4, ...}. There...