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| In [[geometry]], a '''Schwarz triangle''', named after [[Hermann Schwarz]], is a [[spherical triangle]] that can be used to [[tessellation|tile]] a [[sphere]], possibly overlapping, through reflections in its edges. They were classified in {{Harv|Schwarz|1873}}.
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| 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.
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| A Schwarz triangle is represented by three rational numbers (''p'' ''q'' ''r'') each representing the angle at a vertex. The value ''n/d'' means the vertex angle is ''d''/''n'' of the half-circle. "2" means a right triangle. In case 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 3 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 object]]s.
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| == Solution space ==
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| A fundamental domain triangle, (''p'' ''q'' ''r''), can exist in different space depending on this constraint:
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| : <math>
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| \begin{align}
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| \frac 1 p + \frac 1 q + \frac 1 r & > 1 \text{ : Sphere} \\[8pt]
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| \frac 1 p + \frac 1 q + \frac 1 r & = 1 \text{ : Euclidean plane} \\[8pt]
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| \frac 1 p + \frac 1 q + \frac 1 r & < 1 \text{ : Hyperbolic plane.}
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| \end{align}
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| </math>
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| == Graphical representation ==
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| A '''Schwarz triangle''' is represented graphically by a [[Complete graph|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]].
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| {| class=wikitable
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| |[[File:Schwarz triangle on sphere.png|280px]]<BR>Schwarz triangle (''p'' ''q'' ''r'') on sphere
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| |[[File:Schwarz triangle graph.png]]<BR>Schwarz triangle graph
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| |}
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| 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.
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| 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'']×[].
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| == A list of Schwarz triangles ==
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| === Möbius triangles for the sphere ===
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| {| class=wikitable align=right
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| ![[File:Sphere symmetry group d2h.png|120px]]<BR>(2 2 2) or [2,2]
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| ![[File:Sphere symmetry group d3h.png|120px]]<BR>(3 2 2) or [3,2]
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| !...
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| |-
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| ![[File:Sphere symmetry group td.png|120px]]<BR>(3 3 2) or [3,3]
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| ![[File:Sphere symmetry group oh.png|120px]]<BR>(4 3 2) or [4,3]
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| ![[File:Sphere symmetry group ih.png|120px]]<BR>(5 3 2) or [5,3]
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| |}
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| Schwarz triangles with whole numbers, also called '''Möbius triangles''', include one 1-parameter family and three [[exceptional object|exceptional]] cases:
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| # [''p'',2] or (''p'' 2 2) – [[Dihedral symmetry]], {{CDD|node|p|node|2|node}}
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| # [3,3] or (3 3 2) – [[Tetrahedral symmetry]], {{CDD|node|3|node|3|node}}
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| # [4,3] or (4 3 2) – [[Octahedral symmetry]], {{CDD|node|4|node|3|node}}
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| # [5,3] or (5 3 2) – [[Icosahedral symmetry]], {{CDD|node|5|node|3|node}}
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| === Schwarz triangles for the sphere by density ===
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| The Schwarz triangles (''p'' ''q'' ''r''), grouped by [[density (polytope)|density]]:
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| {| class=wikitable
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| !Density
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| !Schwarz triangle
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| |-
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| |1||(2 3 3), (2 3 4), (2 3 5), (2 2 ''n'')
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| |-
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| |''d''||(2 2 ''n''/''d'')
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| |-
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| |2||(3/2 3 3), (3/2 4 4), (3/2 5 5), (5/2 3 3)
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| |-
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| |3||(2 3/2 3), (2 5/2 5)
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| |-
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| |4||(3 4/3 4), (3 5/3 5)
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| |-
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| |5||(2 3/2 3/2), (2 3/2 4)
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| |-
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| |6||(3/2 3/2 3/2), (5/2 5/2 5/2), (3/2 3 5), (5/4 5 5)
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| |-
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| |7||(2 3 4/3), (2 3 5/2)
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| |-
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| |8||(3/2 5/2 5)
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| |-
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| |9||(2 5/3 5)
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| |-
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| |10||(3 5/3 5/2), (3 5/4 5)
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| |-
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| |11||(2 3/2 4/3), (2 3/2 5)
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| |-
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| |13||(2 3 5/3)
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| |-
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| |14||(3/2 4/3 4/3), (3/2 5/2 5/2), (3 3 5/4)
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| |-
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| |16||(3 5/4 5/2)
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| |-
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| |17||(2 3/2 5/2)
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| |-
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| |18||(3/2 3 5/3), (5/3 5/3 5/2)
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| |-
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| |19||(2 3 5/4)
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| |-
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| |21||(2 5/4 5/2)
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| |-
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| |22||(3/2 3/2 5/2)
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| |-
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| |23||(2 3/2 5/3)
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| |-
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| |-
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| |26||(3/2 5/3 5/3)
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| |-
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| |27||(2 5/4 5/3)
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| |-
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| |29||(2 3/2 5/4)
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| |-
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| |32||(3/2 5/45/3)
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| |-
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| |34||(3/2 3/2 5/4)
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| |-
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| |38||(3/2 5/4 5/4)
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| |-
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| |42||(5/4 5/4 5/4)
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| |}
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| === Triangles for the Euclidean plane ===
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| {| class="wikitable" align=right
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| |[[File:Tile 3,6.svg|120px]]<BR>(3 3 3)
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| |[[File:Tile V488 bicolor.svg|120px]]<BR>(4 4 2)
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| |[[File:Tile V46b.svg|120px]]<BR>(6 3 2)
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| |}
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| Density 1:
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| #(3 3 3) – 60-60-60 ([[equilateral triangle|equilateral]])
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| #(4 4 2) – [[45-45-90]] (isosceles right)
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| #(6 3 2) – [[30-60-90]]
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| Rational solutions by density:
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| * Density 0: (4 4/3 ∞), (3 3/2 ∞), (6 6/5 ∞)
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| * Density 1: (4/3 4/3 2), (4/3 4 2), (6 3/2 2)
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| * Density 2: (6/5 3 2), (6 6 3/2), (6 6/5 3)
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| === Triangles for the hyperbolic plane ===
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| {| class="wikitable" align=right
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| |- align=center
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| |[[File:Order-3 heptakis heptagonal tiling.png|120px]]<BR>(7 3 2)
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| |[[File:Order-3 octakis octagonal tiling.png|120px]]<BR>(8 3 2)
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| |[[File:Order-4 bisected pentagonal tiling.png|120px]]<BR>(5 4 2)
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| |- align=center
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| |[[File:Uniform dual tiling 433-t012.png|120px]]<BR>(4 3 3)
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| |[[File:Uniform_dual_tiling_443-t012.png|120px]]<BR>(4 4 3)
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| |[[File:H2checkers_iii.png|120px]]<BR>(∞ ∞ ∞)
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| |- align=center
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| |colspan=3|Fundamental domains of (''p'' ''q'' ''r'') triangles
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| |}
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| Density 1:
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| *(2 3 7), (2 3 8), (2 3 9) ... (2 3 ∞)
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| *(2 4 5), (2 4 6), (2 4 7) ... (2 4 ∞)
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| *(2 5 5), (2 5 6), (2 5 7) ... (2 5 ∞)
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| *(2 6 6), (2 6 7), (2 6 8) ... (2 6 ∞)
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| *(3 3 4), (3 3 5), (3 3 6) ... (3 3 ∞)
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| *(3 4 4), (3 4 5), (3 4 6) ... (3 4 ∞)
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| *(3 5 5), (3 5 6), (3 5 7) ... (3 5 ∞)
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| *(3 6 6), (3 6 7), (3 6 8) ... (3 6 ∞)
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| *...
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| *(∞ ∞ ∞)
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| 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 group]]s – maximal groups of isometries of [[Riemann surface]]s. 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]].
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| The (2 3 8) triangle tiles the [[Bolza surface]], a highly symmetric (but not Hurwitz) surface of genus 2.
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| ==See also==
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| * [[Wythoff symbol]]
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| * [[Wythoff construction]]
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| * [[Uniform polyhedron]]
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| * [[Nonconvex uniform polyhedron]]
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| * [[Density (polytope)]]
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| * [[Goursat tetrahedron]]
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| * [[Regular hyperbolic tiling]]
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| * [[Uniform tilings in hyperbolic plane]]
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| == References ==
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| {{reflist}}
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| {{refbegin}}
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| * [[Coxeter]], ''[[Regular Polytopes (book)|Regular Polytopes]]'', Third edition, (1973), Dover edition, ISBN 0-486-61480-8 (Table 3: Schwarz's Triangles)
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| *{{Citation | last1=Schwarz | first1=H. A. | title=Ueber diejenigen Fälle in welchen die Gaussichen hypergeometrische Reihe eine algebraische Function ihres vierten Elementes darstellt | url=http://resolver.sub.uni-goettingen.de/purl?GDZPPN002155206 | year=1873 | volume=75 | journal=[[Journal für die reine und angewandte Mathematik]] | issn=0075-4102 | pages=292–335}} (Note that Coxeter references this as "Zur Theorie der hypergeometrischen Reihe", which is the short title used in the journal page headers).
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| * {{Citation
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| | publisher = CUP Archive
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| | isbn = 978-0-521-22279-2
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| | last = Wenninger
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| | first = Magnus J.
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| | title = Spherical models
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| | chapter = An introduction to the notion of polyhedral density
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| | pages = [http://books.google.com/books?id=Olc5AAAAIAAJ&pg=PA132 132–134]
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| | year = 1979
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| }}
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| {{refend}}
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| == External links ==
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| * {{mathworld | urlname = SchwarzTriangle | title = Schwarz triangle}}
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| * [http://ogre.nu/klitzing/explain/pqr.htm The general Schwarz triangle (p q r) and the generalized incidence matrices of the corresponding polyhedra]
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| [[Category:Spherical trigonometry]]
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| [[Category:Polyhedra]]
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