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.
Solution space A fundamental domain triangle , with vertex angles , , and , can exist in different spaces depending on the value of the sum of the reciprocals of these integers:
This is simply a way of saying that in Euclidean space the interior angles of a triangle sum to , while on a sphere they sum to an angle greater than , and on hyperbolic space they sum to less.
Graphical representation A Schwarz triangle is represented graphically by a triangular graph. Each node represents an edge (mirror) of the Schwarz triangle. Each edge is labeled by a rational value corresponding to the reflection order, being π/vertex angle.
Schwarz triangle (p q r) on sphereSchwarz triangle graph
Order-2 edges represent perpendicular mirrors that can be ignored in this diagram. The Coxeter-Dynkin diagram represents this triangular graph with order-2 edges hidden.
A Coxeter group can be used for a simpler notation, as (p q r) for cyclic graphs, and (p q 2) = [p,q] for (right triangles), and (p 2 2) = [p]×[].
A list of Schwarz triangles
Möbius triangles for the sphere (2 2 2) or [2,2](3 2 2) or [3,2]...(3 3 2) or [3,3](4 3 2) or [4,3](5 3 2) or [5,3]
Schwarz triangles with whole numbers, also called Möbius triangles, include one 1-parameter family and three exceptional cases:
[p,2] or (p 2 2) – Dihedral symmetry, [3,3] or (3 3 2) – Tetrahedral symmetry, [4,3] or (4 3 2) – Octahedral symmetry, [5,3] or (5 3 2) – Icosahedral symmetry,
Schwarz triangles for the sphere by density The Schwarz triangles (p q r), grouped by density:
DensityDihedralTetrahedralOctahedralIcosahedrald(2 2 n/d) 1 (2 3 3) (2 3 4) (2 3 5)2 (3/2 3 3) (3/2 4 4) (3/2 5 5), (5/2 3 3)3 (2 3/2 3) (2 5/2 5)4 (3 4/3 4) (3 5/3 5)5 (2 3/2 3/2) (2 3/2 4)6 (3/2 3/2 3/2) (5/2 5/2 5/2), (3/2 3 5), (5/4 5 5)7 (2 3 4/3) (2 3 5/2)8 (3/2 5/2 5)9 (2 5/3 5)10 (3 5/3 5/2), (3 5/4 5)11 (2 3/2 4/3) (2 3/2 5)13 (2 3 5/3)14 (3/2 4/3 4/3) (3/2 5/2 5/2), (3 3 5/4)16 (3 5/4 5/2)17 (2 3/2 5/2)18 (3/2 3 5/3), (5/3 5/3 5/2)19 (2 3 5/4)21 (2 5/4 5/2)22 (3/2 3/2 5/2)23 (2 3/2 5/3)26 (3/2 5/3 5/3)27 (2 5/4 5/3)29 (2 3/2 5/4)32 (3/2 5/4 5/3)34 (3/2 3/2 5/4)38 (3/2 5/4 5/4)42 (5/4 5/4 5/4)
Triangles for the Euclidean plane (3 3 3)(4 4 2)(6 3 2)
Density 1:
(3 3 3) – 60-60-60 (equilateral), (4 4 2) – 45-45-90 (isosceles right), (6 3 2) – 30-60-90,
Density 2:
(6 6 3/2) - 120-30-30 triangle
Density ∞:
(4 4/3 ∞) (3 3/2 ∞) (6 6/5 ∞)
Triangles for the hyperbolic plane (7 3 2)(8 3 2)(5 4 2)(4 3 3)(4 4 3)(∞ ∞ ∞)Fundamental domains of (p q r) triangles Density 1:
(2 3 7), (2 3 8), (2 3 9) ... (2 3 ∞) (2 4 5), (2 4 6), (2 4 7) ... (2 4 ∞) (2 5 5), (2 5 6), (2 5 7) ... (2 5 ∞) (2 6 6), (2 6 7), (2 6 8) ... (2 6 ∞) (3 3 4), (3 3 5), (3 3 6) ... (3 3 ∞) (3 4 4), (3 4 5), (3 4 6) ... (3 4 ∞) (3 5 5), (3 5 6), (3 5 7) ... (3 5 ∞) (3 6 6), (3 6 7), (3 6 8) ... (3 6 ∞) ... (∞ ∞ ∞)
Density 2:
(3/2 7 7), (3/2 8 8), (3/2 9 9) ... (3/2 ∞ ∞) (5/2 4 4), (5/2 5 5), (5/2 6 6) ... (5/2 ∞ ∞) (7/2 3 3), (7/2 4 4), (7/2 5 5) ... (7/2 ∞ ∞) (9/2 3 3), (9/2 4 4), (9/2 5 5) ... (9/2 ∞ ∞) ...
Density 3:
(2 7/2 7), (2 9/2 9), (2 11/2 11) ...
Density 4:
(7/3 3 7), (8/3 3 8), (3 10/3 10), (3 11/3 11) ...
Density 6:
(7/4 7 7), (9/4 9 9), (11/4 11 11) ... (7/2 7/2 7/2), (9/2 9/2 9/2), ...
Density 10:
(3 7/2 7)
The (2 3 7) Schwarz triangle is the smallest hyperbolic Schwarz triangle, and as such is of particular interest. Its triangle group (or more precisely the index 2 von Dyck group of orientation-preserving isometries) is the (2,3,7) triangle group, which is the universal group for all Hurwitz groups – maximal groups of isometries of Riemann surfaces. All Hurwitz groups are quotients of the (2,3,7) triangle group, and all Hurwitz surfaces are tiled by the (2,3,7) Schwarz triangle. The smallest Hurwitz group is the simple group of order 168, the second smallest non-abelian simple group, which is isomorphic to PSL(2,7), and the associated Hurwitz surface (of genus 3) is the Klein quartic.
The (2 3 8) triangle tiles the Bolza surface, a highly symmetric (but not Hurwitz) surface of genus 2.
The triangles with one noninteger angle, listed above, were first classified by Anthony W. Knapp in.A. W. Knapp, Doubly generated Fuchsian groups, Michigan Mathematical Journal 15 (1968), no. 3, 289–304 A list of triangles with multiple noninteger angles is given in.Klimenko and Sakuma, Two-generator discrete subgroups of Isom( H 2 ) containing orientation-reversing elements, Geometriae Dedicata October 1998, Volume 72, Issue 3, pp 247-282
Tessellation by Schwarz triangles 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.
Triangles without cusps
Suppose that the hyperbolic triangle Δ has angles /a, /b and /c with a, b, c integers greater than 1. The hyperbolic area of Δ equals − /a − /b − /c, so that
The construction of a tessellation will first be carried out for the case when a, b and c are greater than 2.
The original triangle Δ gives a convex polygon P1 with 3 vertices. At each of the three vertices the triangle can be successively reflected through edges emanating from the vertices to produce 2m copies of the triangle where the angle at the vertex is /m. The triangles do not overlap except at the edges, half of them have their orientation reversed and they fit together to tile a neighborhood of the point. The union of these new triangles together with the original triangle form a connected shape P2. It is made up of triangles which only intersect in edges or vertices, forms a convex polygon with all angles less than or equal to and each side being the edge of a reflected triangle. In the case when an angle of Δ equals /3, a vertex of P2 will have an interior angle of , but this does not affect the convexity of P2. Even in this degenerate case when an angle of arises, the two collinear edges are still considered as distinct for the purposes of the construction.
The construction of P2 can be understood more clearly by noting that some triangles or tiles are added twice, the three which have a side in common with the original triangle. The rest have only a vertex in common. A more systematic way of performing the tiling is first to add a tile to each side (the reflection of the triangle in that edge) and then fill in the gaps at each vertex. This results in a total of 3 + (2a − 3) + (2b − 3) + (2c − 3) = 2(a + b + c) − 6 new triangles. The new vertices are of two types. Those which are vertices of the triangles attached to sides of the original triangle, which are connected to 2 vertices of Δ. Each of these lie in three new triangles which intersect at that vertex. The remainder are connected to a unique vertex of Δ and belong to two new triangles which have a common edge. Thus there are 3 + (2a − 4) + (2b − 4) + (2c − 4) = 2(a + b + c) − 9 new vertices. By construction there is no overlapping. To see that P2 is convex, it suffices to see that the angle between sides meeting at a new vertex make an angle less than or equal to . But the new vertices lies in two or three new triangles, which meet at that vertex, so the angle at that vertex is no greater than 2/3 or , as required.
This process can be repeated for P2 to get P3 by first adding tiles to each edge of P2 and then filling in the tiles round each vertex of P2. Then the process can be repeated from P3, to get P4 and so on, successively producing Pn from Pn − 1. It can be checked inductively that these are all convex polygons, with non-overlapping tiles. Indeed, as in the first step of the process there are two types of tile in building Pn from Pn − 1, those attached to an edge of Pn − 1 and those attached to a single vertex. Similarly there are two types of vertex, one in which two new tiles meet and those in which three tiles meet. So provided that no tiles overlap, the previous argument shows that angles at vertices are no greater than and hence that Pn is a convex polygon.
It therefore has to be verified that in constructing Pn from Pn − 1:
To prove (a), note that by convexity, the polygon Pn − 1 is the intersection of the convex half-spaces defined by the full circular arcs defining its boundary. Thus at a given vertex of Pn − 1 there are two such circular arcs defining two sectors: one sector contains the interior of Pn − 1, the other contains the interiors of the new triangles added around the given vertex. This can be visualized by using a Möbius transformation to map the upper half plane to the unit disk and the vertex to the origin; the interior of the polygon and each of the new triangles lie in different sectors of the unit disk. Thus (a) is proved.
Before proving (c) and (b), a Möbius transformation can be applied to map the upper half plane to the unit disk and a fixed point in the interior of Δ to the origin.