Шарды мозаикалауға қолданылатын сфералық үшбұрыштар

Spherical triangle that can be used to tile a sphere

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, through reflections in its edges. They were classified in
These can be defined more generally as tessellations of the sphere, the Euclidean plane, or the hyperbolic plane. Each Schwarz triangle on a sphere defines a finite group, while on the Euclidean or hyperbolic plane they define an infinite group. A Schwarz triangle is represented by three rational numbers each representing the angle at a vertex. The value means the vertex angle is of the half circle. "2" means a right triangle. When these are whole numbers, the triangle is called a Möbius triangle, and corresponds to a non overlapping tiling, and the symmetry group is called a triangle group. In the sphere there are three Möbius triangles plus one one parameter family; in the plane there are three Möbius triangles, while in hyperbolic space there is a three parameter family of Möbius triangles, and no exceptional objects.

In this section tessellations of the hyperbolic upper half plane by Schwarz triangles will be discussed using elementary methods. For triangles without "cusps"—angles equal to zero or equivalently vertices on the real axis—the elementary approach of will be followed. For triangles with one or two cusps, elementary arguments of , simplifying the approach of , will be used: in the case of a Schwarz triangle with one angle zero and another a right angle, the orientation preserving subgroup of the reflection group of the triangle is a Hecke group. For an ideal triangle in which all angles are zero, so that all vertices lie on the real axis, the existence of the tessellation will be established by relating it to the Farey series described in and In this case the tessellation can be considered as that associated with three touching circles on the Riemann sphere, a limiting case of configurations associated with three disjoint non nested circles and their reflection groups, the so called "Schottky groups", described in detail in Alternatively—by dividing the ideal triangle into six triangles with angles 0, /2 and /3—the tessellation by ideal triangles can be understood in terms of tessellations by triangles with one or two cusps.

In this subsection the approach of Carl Ludwig Siegel to the tessellation theorem for triangles is outlined. Siegel's less elementary approach does not use convexity, instead relying on the theory of Riemann surfaces, covering spaces and a version of the monodromy theorem for coverings. It has been generalized to give proofs of the more general Poincaré polygon theorem. (Note that the special case of tiling by regular n gons with interior angles 2/n is an immediate consequence of the tessellation by Schwarz triangles with angles /n, /n and /2.) Let Γ be the free product Z2 ∗ Z2 ∗ Z2. If Δ = ABC is a Schwarz triangle with angles /a, /b and /c, where a, b, c ≥ 2, then there is a natural map of Γ onto the group generated by reflections in the sides of Δ. Elements of Γ are described by a product of the three generators where no two adjacent generators are equal. At the vertices A, B and C the product of reflections in the sides meeting at the vertex define rotations by angles 2/a, 2/b and 2/c; Let gA, gB and gC be the corresponding products of generators of Γ = Z2 ∗ Z2 ∗ Z2. Let Γ0 be the normal subgroup of index 2 of Γ, consisting of elements that are the product of an even number of generators; and let Γ1 be the normal subgroup of Γ generated by (gA)a, (gB)b and (gC)c. These act trivially on Δ. Let = Γ/Γ1 and 0 = Γ0/Γ1. The disjoint union of copies of indexed by elements of with edge identifications has the natural structure of a Riemann surface Σ. At an interior point of a triangle there is an obvious chart. As a point of the interior of an edge the chart is obtained by reflecting the triangle across the edge. At a vertex of a triangle with interior angle /n, the chart is obtained from the 2n copies of the triangle obtained by reflecting it successively around that vertex. The group acts by deck transformations of Σ, with elements in 0 acting as holomorphic mappings and elements not in 0 acting as antiholomorphic mappings. There is a natural map P of Σ into the hyperbolic plane. The interior of the triangle with label g in is taken onto g(Δ), edges are taken to edges and vertices to vertices. It is also easy to verify that a neighbourhood of an interior point of an edge is taken into a neighbourhood of the image; and similarly for vertices. Thus P is locally a homeomorphism and so takes open sets to open sets. The image P(Σ), i. e. the union of the translates g , is therefore an open subset of the upper half plane. On the other hand, this set is also closed. Indeed, if a point is sufficiently close to it must be in a translate of Indeed, a neighbourhood of each vertex is filled out the reflections of and if a point lies outside these three neighbourhoods but is still close to it must lie on the three reflections of in its sides. Thus there is δ > 0 such that if z lies within a distance less than δ from , then z lies in a translate of Since the hyperbolic distance is invariant, it follows that if z lies within a distance less than δ from Γ it actually lies in Γ , so this union is closed. By connectivity it follows that P(Σ) is the whole upper half plane. On the other hand, P is a local homeomorphism, so a covering map. Since the upper half plane is simply connected, it follows that P is one one and hence the translates of Δ tessellate the upper half plane. This is a consequence of the following version of the monodromy theorem for coverings of Riemann surfaces: if Q is a covering map between Riemann surfaces Σ1 and Σ2, then any path in Σ2 can be lifted to a path in Σ1 and any two homotopic paths with the same end points lift to homotopic paths
with the same end points; an immediate corollary is that if Σ2 is simply connected, Q must be a homeomorphism. To apply this, let Σ1 = Σ, let Σ2 be the upper half plane and let Q = P. By the corollary of the monodromy theorem, P must be one one. It also follows that g(Δ) = Δ if and only if g lies in Γ1, so that the homomorphism of 0 into the Möbius group is faithful.

gave a general proof of Poincaré's polygon theorem in hyperbolic space; a similar proof was given in Specializing to the hyperbolic plane and Schwarz triangles, this can be used to give a modern approach for establishing that the existence of Schwarz triangle tessellations, as described in and The Swiss mathematicians and Haefliger have provided an introductory account, taking geometric group theory as their starting point.