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| {{about|a geometric concept|Weihai, China-based tire manufacturing company|Triangle Group (company)}}
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| In [[mathematics]], a '''triangle group''' is a [[group (mathematics)|group]] that can be realized geometrically by sequences of [[reflection (mathematics)|reflections]] across the sides of a [[triangle]]. The triangle can be an ordinary [[Euclidean geometry|Euclidean]] triangle, a [[Schwarz triangle|triangle on the sphere]], or a [[hyperbolic triangle]]. Each triangle group is the [[symmetry group]] of a [[tessellation|tiling]] of the [[Euclidean plane]], the [[sphere]], or the [[Hyperbolic space|hyperbolic plane]] by [[congruence (geometry)|congruent]] triangles, a [[fundamental domain]] for the action, called a [[Möbius triangle]].
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| ==Definition==
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| Let ''l'', ''m'', ''n'''' be [[integer]]s greater than or equal to 2. A '''triangle group''' Δ(''l'',''m'',''n'') is a group of motions of the Euclidean plane, the two-dimensional sphere, the real projective plane, or the hyperbolic plane generated by the [[Reflection (mathematics)|reflection]]s in the sides of a [[triangle]] with angles π/''l'', π/''m'' and π/''n'' (measured in [[radian]]s). The product of the reflections in two adjacent sides is a [[rotation]] by the angle which is twice the angle between those sides, 2π/''l'', 2π/''m'' and 2π/''n'' Therefore, if the generating reflections are labeled ''a'', ''b'', ''c'' and the angles between them in the cyclic order are as given above, then the following relations hold:
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| # <math> a^{2}=b^{2}=c^{2}=1 </math>
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| # <math> (ab)^{l}=(bc)^{n}=(ca)^{m}=1. </math>
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| It is a theorem that all other relations between ''a, b, c'' are consequences of these relations and that Δ(''l,m,n'') is a [[discrete group]] of motions of the corresponding space. Thus a triangle group is a [[reflection group]] that admits a [[group presentation]] | |
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| :<math> \Delta(l,m,n) = \langle a,b,c \mid a^{2} = b^{2} = c^{2} = (ab)^{l} = (bc)^{n} = (ca)^{m} = 1 \rangle. </math>
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| An abstract group with this presentation is a [[Coxeter group]] with three generators.
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| ==Classification==
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| Given any natural numbers ''l'', ''m'', ''n'' > 1 exactly one of the classical two-dimensional geometries (Euclidean, spherical, or hyperbolic) admits a triangle with the angles (π/l, π/m, π/n), and the space is tiled by reflections of the triangle. The sum of the angles of the triangle determines the type of the geometry by the [[Gauss–Bonnet theorem]]: it is Euclidean if the angle sum is exactly π, spherical if it exceeds π and hyperbolic if it is strictly smaller than π. Moreover, any two triangles with the given angles are congruent. Each triangle group determines a tiling, which is conventionally colored in two colors, so that any two adjacent tiles have opposite colors.
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| In terms of the numbers ''l'', ''m'', ''n'' > 1 there are the following possibilities.
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| ===The Euclidean case===
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| <math>\frac{1}{l}+\frac{1}{m}+\frac{1}{n}=1.</math>
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| The triangle group is the infinite [[symmetry group]] of a certain [[tessellation]] (or tiling) of the Euclidean plane by triangles whose angles add up to π (or 180°). Up to permutations, the triple (''l'', ''m'', ''n'') is one of the triples (2,3,6), (2,4,4), (3,3,3). The corresponding triangle groups are instances of [[wallpaper group]]s.
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| {| class=wikitable
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| !(2,3,6)
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| !(2,4,4)
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| !(3,3,3)
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| |-
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| |[[File:Tiling Dual Semiregular V4-6-12 Bisected Hexagonal.svg|240px]]
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| |[[File:Tile_V488_bicolor.svg|240px]]
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| |[[File:Tiling Regular 3-6 Triangular.svg|240px]]
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| |-
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| ![[bisected hexagonal tiling]]
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| ![[tetrakis square tiling]]
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| ![[triangular tiling]]
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| |-
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| |colspan=3|More detailed diagrams, labeling the vertices and showing how reflection operates:
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| |- align=center
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| |[[File:Wallpaper group diagram p6m.svg|240px]]
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| |[[File:Wallpaper group diagram p4m square.svg|150px]]
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| |[[File:Wallpaper group diagram p3m1.svg|240px]]
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| |}
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| ===The spherical case===
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| :<math>\frac{1}{l}+\frac{1}{m}+\frac{1}{n}>1.</math>
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| The triangle group is the finite symmetry group of a tiling of a unit sphere by spherical triangles, or [[Möbius triangle]]s, whose angles add up to a number greater than π. Up to permutations, the triple (''l'',''m'',''n'') has the form (2,3,3), (2,3,4), (2,3,5), or (2,2,''n''), ''n'' > 1. Spherical triangle groups can be identified with the symmetry groups of [[platonic solid|regular polyhedra]] in the three-dimensional Euclidean space: Δ(2,3,3) corresponds to the [[tetrahedron]], Δ(2,3,4) to both the [[cube]] and the [[octahedron]] (which have the same symmetry group), Δ(2,3,5) to both the [[dodecahedron]] and the [[icosahedron]]. The groups Δ(2,2,''n''), ''n'' > 1 of [[dihedral symmetry]] can be interpreted as the symmetry groups of the family of [[dihedron]]s, which are degenerate solids formed by two identical [[regular polygon|regular ''n''-gons]] joined together, or dually [[hosohedron]]s, which are formed by joining ''n'' [[digon]]s together at two vertices.
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| The [[spherical tiling]] corresponding to a regular polyhedron is obtained by forming the [[barycentric subdivision]] of the polyhedron and projecting the resulting points and lines onto the circumscribed sphere. In the case of the tetrahedron, there are four faces and each face is an equilateral triangle that is subdivided into 6 smaller pieces by the medians intersecting in the center. The resulting tesselation has 4 × 6=24 spherical triangles (it is the spherical [[disdyakis cube]]).
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| These groups are finite, which corresponds to the compactness of the sphere – areas of discs in the sphere initially grow in terms of radius, but eventually cover the entire sphere.
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| The triangular tilings are depicted below:
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| {| class="wikitable"
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| !(2,2,2)
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| !(2,2,3)
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| !(2,2,4)
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| !(2,2,5)
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| !(2,2,6)
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| !(2,2,n)
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| |- align=center
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| |[[Image:Spherical square bipyramid2.png|80px]]
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| |[[Image:Spherical hexagonal bipyramid2.png|80px]]
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| |[[Image:Spherical octagonal bipyramid2.png|80px]]
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| |[[Image:Spherical decagonal bipyramid2.png|80px]]
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| |[[Image:Spherical dodecagonal bipyramid2.png|80px]]
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| |-
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| !colspan=2|(2,3,3)
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| !colspan=2|(2,3,4)
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| !colspan=2|(2,3,5)
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| |- align=center
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| |colspan=2|[[Image:Tetrahedral reflection domains.png|150px]]
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| |colspan=2|[[Image:Octahedral reflection domains.png|150px]]
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| |colspan=2|[[Image:Icosahedral reflection domains.png|150px]]
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| |}
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| Spherical tilings corresponding to the octahedron and the icosahedron and dihedral spherical tilings with even ''n'' are [[central symmetry|centrally symmetric]]. Hence each of them determines a tiling of the real projective plane, an '''[[elliptic tiling]]'''. Its symmetry group is the quotient of the spherical triangle group by the [[reflection through the origin]] (-''I''), which is a central element of order 2. Since the projective plane is a model of [[elliptic geometry]], such groups are called ''elliptic'' triangle groups.<ref>{{Harv|Magnus|1974}}</ref>
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| ===The hyperbolic case===
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| :<math>\frac{1}{l}+\frac{1}{m}+\frac{1}{n}<1.</math>
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| The triangle group is the infinite symmetry group of a [[Uniform tilings in hyperbolic plane|tiling of the hyperbolic plane]] by hyperbolic triangles whose angles add up to a number less than π. All triples not already listed represent tilings of the hyperbolic plane. For example, the triple (2,3,7) produces the [[(2,3,7) triangle group]]. There are infinitely many such groups; the tilings associated with some small values:
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| ==== Hyperbolic plane ====
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| {| class="wikitable"
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| |+ [[Poincaré disk model]] of fundamental domain triangles
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| |-
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| !colspan=5|Example right triangles (2 p q)
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| |- align=center
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| |[[File:H2checkers_237.png|100px]]<BR>[[(2,3,7) triangle group|(2 3 7)]]
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| |[[File:H2checkers_238.png|100px]]<BR>(2 3 8)
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| |[[File:Hyperbolic domains 932.png|100px]]<BR>(2 3 9)
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| |[[File:H2checkers_23i.png|100px]]<BR>(2 3 ∞)
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| |- align=center
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| |[[File:H2checkers_245.png|100px]]<BR>(2 4 5)
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| |[[File:H2checkers_246.png|100px]]<BR>(2 4 6)
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| |[[File:H2checkers_247.png|100px]]<BR>(2 4 7)
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| |[[File:H2checkers_248.png|100px]]<BR>(2 4 8)
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| |[[File:H2checkers_24i.png|100px]]<BR>(2 4 ∞)
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| |- align=center
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| |[[File:H2checkers_255.png|100px]]<BR>(2 5 5)
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| |[[File:H2checkers_256.png|100px]]<BR>(2 5 6)
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| |[[File:H2checkers_257.png|100px]]<BR>(2 5 7)
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| |[[File:H2checkers_266.png|100px]]<BR>(2 6 6)
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| |[[File:H2checkers_2ii.png|100px]]<BR>(2 ∞ ∞)
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| |- align=center
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| !colspan=5|Example general triangles (p q r)
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| |- align=center
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| |[[File:H2checkers_334.png|100px]]<BR>(3 3 4)
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| |[[File:H2checkers_335.png|100px]]<BR>(3 3 5)
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| |[[File:H2checkers_336.png|100px]]<BR>(3 3 6)
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| |[[File:H2checkers_337.png|100px]]<BR>(3 3 7)
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| |[[File:H2checkers 33i.png|100px]]<BR>(3 3 ∞)
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| |- align=center
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| |[[File:H2checkers_344.png|100px]]<BR>(3 4 4)
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| |[[File:H2checkers_366.png|100px]]<BR>(3 6 6)
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| |[[File:H2checkers 3ii.png|100px]]<BR>(3 ∞ ∞)
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| |[[File:H2checkers_666.png|100px]]<BR>(6 6 6)
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| |[[File:H2checkers iii.png|100px]]<BR>(∞ ∞ ∞)
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| |}
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| Hyperbolic triangle groups are examples of [[non-Euclidean crystallographic group]] and have been generalized in the theory of [[Mikhail Gromov (mathematician)|Gromov]] [[hyperbolic group]]s.
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| == von Dyck groups ==
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| Denote by ''D''(''l'',''m'',''n'') the [[subgroup]] of [[Index of a subgroup|index]] 2 in ''Δ(l,m,n)'' generated by words of even length in the generators. Such subgroups are sometimes referred to as "ordinary" triangle groups<ref>{{Harv |Gross |Tucker |2001}}</ref> or '''von Dyck groups''', after [[Walther von Dyck]]. For spherical, Euclidean, and hyperbolic triangles, these correspond to the elements of the group that preserve the [[Orientation (vector space)|orientation]] of the triangle – the group of rotations. For projective (elliptic) triangles, they cannot be so interpreted, as the projective plane is non-orientable, so there is no notion of "orientation-preserving". The reflections are however ''locally'' orientation-reversing (and every manifold is locally orientable, because locally Euclidean): they fix a line and at each point in the line are a reflection across the line.<ref>{{Harv|Magnus|1974|loc=p. 65}}</ref>
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| The groups ''D''(''l'',''m'',''n'') is defined by the following presentation:
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| :<math>D(l,m,n)=\langle x,y \mid x^l,y^m,(xy)^n\rangle.</math>
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| In terms of the generators above, these are ''x = ab, y = ca, yx = cb''. Geometrically, the three elements ''x'', ''y'', ''xy'' correspond to rotations by 2π/''l'', 2π/''m'' and 2π/''n'' about the three vertices of the triangle.
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| Note that ''D''(''l'',''m'',''n'') ≅ ''D''(''m'',''l'',''n'') ≅ ''D''(''n'',''m'',''l''), so ''D''(''l'',''m'',''n'') is independent of the order of the ''l'',''m'',''n''.
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| A hyperbolic von Dyck group is a [[Fuchsian group]], a discrete group consisting of orientation-preserving isometries of the hyperbolic plane.
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| == Overlapping tilings ==
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| {{further|Schwarz triangle}}
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| Triangle groups preserve a tiling by triangles, namely a [[fundamental domain]] for the action (the triangle defined by the lines of reflection), called a [[Möbius triangle]], and are given by a triple of ''integers,'' (''l'',''m'',''n''), – integers correspond to (2''l'',2''m'',2''n'') triangles coming together at a vertex. There are also tilings by overlapping triangles, which correspond to [[Schwarz triangle]]s with ''rational'' numbers (''l''/''a'',''m''/''b'',''n''/''c''), where the denominators are [[coprime]] to the numerators. This corresponds to edges meeting at angles of ''a''π/''l'' (resp.), which corresponds to a rotation of 2''a''π/''l'' (resp.), which has order ''l'' and is thus identical as an abstract group element, but distinct when represented by a reflection.
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| For example, the Schwarz triangle (2 3 3) yields a [[density (polytope)|density]] 1 tiling of the sphere, while the triangle (2 3/2 3) yields a density 3 tiling of the sphere, but with the same abstract group. These symmetries of overlapping tilings are not considered triangle groups.
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| ==History==
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| Triangle groups date at least to the presentation of the [[icosahedral group]] as the (rotational) (2,3,5) triangle group by [[William Rowan Hamilton]] in 1856, in his paper on [[icosian calculus]].<ref>{{citation
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| |title=Memorandum respecting a new System of Roots of Unity
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| |author=Sir William Rowan Hamilton
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| |author-link=William Rowan Hamilton
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| |url=http://www.maths.tcd.ie/pub/HistMath/People/Hamilton/Icosian/NewSys.pdf
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| |journal=[[Philosophical Magazine]]
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| |volume=12
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| |year=1856
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| |pages=446
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| }}</ref>
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| ==Applications==
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| {{external media
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| | video1 = [http://www.xs4all.nl/~westy31/Geometry/vlib.gif Warped modular tiling]<ref name="westendorp">[http://www.xs4all.nl/~westy31/Geometry/Geometry.html#Modular Platonic tilings of Riemann surfaces: The Modular Group], [http://www.xs4all.nl/~westy31/ Gerard Westendorp]</ref> – visualization of the map (2,3,∞) → (2,3,7) by morphing the associated tilings.
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| }}
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| Triangle groups arise in [[arithmetic geometry]]. The [[modular group]] is generated by two elements, ''S'' and ''T'', subject to the relations ''S² = (ST)³ = 1'' (no relation on ''T''), is the rotational triangle group (2,3,∞) and maps onto all triangle groups (2,3,''n'') by adding the relation ''T''<sup>''n''</sup> = 1. More generally, the [[Hecke group]] ''H''<sub>''q''</sub> is by two elements, ''S'' and ''T'', subject to the relations ''S''<sup>2</sup> = (''ST'')<sup>''q''</sup> = 1 (no relation on ''T''), is the rotational triangle group (2,''q'',∞), and maps onto all triangle groups (2,''q'',''n'') by adding the relation ''T''<sup>''n''</sup> = 1 the modular group is the Hecke group ''H''<sub>3</sub>. In [[Alexander Grothendieck|Grothendieck]]'s theory of [[dessins d'enfants]], a [[Belyi function]] gives rise to a tessellation of a [[Riemann surface]] by reflection domains of a triangle group.
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| All 26 [[sporadic group]]s are quotients of triangle groups,<ref>{{Harv|Wilson|2001|loc = Table 2, p. 7}}</ref> of which 12 are Hurwitz groups (quotients of the (2,3,7) group).
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| ==See also==
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| * [[Schwarz triangle]]
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| * The [[Schwarz triangle map]] is a map of triangles to the [[upper half-plane]].
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| * [[Geometric group theory]]
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| ==References==
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| {{reflist}}
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| {{refimprove|date=April 2010}}
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| {{refbegin}}
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| * {{Citation
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| | publisher = [[Academic Press]]
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| | isbn = 978-0-12-465450-1
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| | last = Magnus
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| | first = Wilhelm
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| | authorlink = Wilhelm Magnus
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| | title = Noneuclidean tesselations and their groups
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| | chapter = II. Discontinuous groups and triangle tessellations
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| | pages = [http://books.google.com/books?id=iLkzandfCc8C&pg=PA52 52–106]
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| | year = 1974
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| }}
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| * {{Citation
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| | publisher = Courier Dover Publications
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| | isbn = 978-0-486-41741-7
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| | last1 = Gross
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| | first1 = Jonathan L.
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| | first2 = Thomas W. | last2 = Tucker
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| | title = Topological graph theory
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| | year = 2001
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| | chapter = 6.2.8 Triangle Groups
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| | pages = [http://books.google.com/books?id=mrv9OJVdy_cC&pg=RA1-PA279 279–281]
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| }}
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| * {{cite doi|10.1515/jgth.2001.027}}
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| {{refend}}
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| ==External links==
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| * Robert Dawson [http://cs.smu.ca/~dawson/images4.html Some spherical tilings] (undated, earlier than 2004) ''(Shows a number of interesting sphere tilings, most of which are not triangle group tilings.)''
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| * Elizabeth r chen [http://www-personal.umich.edu/~bethchen/macosx/trigroup.html triangle groups] (2010) desktop background pictures
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| {{PlanetMath attribution|id=5925|title=Triangle groups}}
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| [[Category:Finite groups]]
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| [[Category:Polyhedra]]
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| [[Category:Tessellation]]
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| [[Category:Spherical trigonometry]]
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| [[Category:Euclidean geometry]]
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| [[Category:Hyperbolic geometry]]
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| [[Category:Properties of groups]]
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| [[Category:Coxeter groups]]
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| [[Category:Geometric group theory]]
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