Triangle group

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In mathematics, the triangle groups are groups that can be realized geometrically by sequences of reflections across the sides of certain triangles. Each triangle group represents symmetries of a tiling by congruent triangles. The triangle can be an ordinary Euclidean triangle, or a triangle in the sphere, or in the hyperbolic plane.

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[edit] Definition

The triangle group Δ(l,m,n) is a group of motions of the Euclidean plane, the two-dimensional sphere, or the hyperbolic plane. Given a triangle with the angles \frac{\pi}{l}, \frac{\pi}{m}, and \frac{\pi}{n}, the corresponding group is generated by the reflections in the sides of the triangle. The product of the reflections in two adjacent sides is a rotation by the angle which the double of the angle between the sides. 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:

a^2=b^2=c^2=1, \quad (ab)^l=(bc)^n=(ca)^m=1.

It is a theorem that all other relations between a,b,c are consequences of these relations. An abstract triangle group can be defined by the group presentation

\Delta(l,m,n)=\langle a,b,c \mid a^2,b^2,c^2,(ab)^l,(bc)^n,(ca)^m\rangle

where l,m,n are integers greater than or equal to 2. Triangle groups are Coxeter groups with three generators.

[edit] Classification

Given any natural numbers l, m, n\geq 2, there is exactly one two-dimensional geometry (Euclidean, spherical, or hyperbolic) which admits a triangle with the angles \frac{\pi}{l}, \frac{\pi}{m}, and \frac{\pi}{n}. Moreover, any two such triangles are congruent. The type of the geometry is determined by the sum of the angles of the triangle: Euclidean if it is exactly π, spherical if it exceeds π, and hyperbolic if it is strictly smaller than π.

In terms of the numbers l, m, n\geq 2, there are the following possibilities:

  • The Euclidean case:
\frac{1}{l}+\frac{1}{m}+\frac{1}{n}=1.
The triangle group is the infinite symmetry group of a certain tiling of the Euclidean plane by ordinary triangles whose angles add up to π. 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 groups.
  • The spherical case:
\frac{1}{l}+\frac{1}{m}+\frac{1}{n}>1.
The triangle group is the finite symmetry group of a tiling of the sphere by spherical triangles or Schwarz triangles, 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). Spherical triangle groups can be identified with the symmetry groups of regular polyhedra: Δ(2,3,3) corresponds to the tetrahedron, Δ(2,3,4) to the cube and octahedron (which have the same symmetry group), Δ(2,3,5) to the dodecahedron and icosahedron. The groups \Delta(2,2,n), n\geq 2 can be interpreted as the symmetry groups of a family of "degenerate solids" formed by two identical regular n-gons stuck together. To obtain the spherical tiling, start with a regular polyhedron, make the barycentric subdivision, and project the resulting points and lines onto the circumscribed sphere. For example, in the case of the tetrahedron, this results in 6*4=24 spherical triangles.
  • The hyperbolic case:
\frac{1}{l}+\frac{1}{m}+\frac{1}{n}<1.
The triangle group is the infinite symmetry group of a tiling of the hyperbolic plane by triangles whose angles add up to a number less than π. All triples not already listed, such as (l,m,n) = (2,3,7), represent tilings of the hyperbolic plane.

[edit] von Dyck groups

Denote by D(l,m,n) the subgroup of index 2 in Δ(l,m,n), corresponding to preservation of orientation of the triangle. Such subgroups are sometimes referred to as von Dyck groups.

The D(l,m,n) are defined by the following presentation:

D(l,m,n)=\langle x,y \mid x^l,y^m,(xy)^n\rangle

Note that

D(l,m,n)\cong D(m,l,n)\cong D(n,m,l),

so D(l,m,n) is independent of the order of the l,m,n.

[edit] See also

[edit] References

  • Robert Dawson Some spherical tilings (undated, earlier than 2004) (Shows a number of interesting sphere tilings, most of which are not triangle group tilings.)

This article incorporates material from Triangle groups on PlanetMath, which is licensed under the GFDL.