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.

[edit] Definition

A triangle group is 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.

A group with this presentation corresponds to a triangle whose angles are $\frac{\pi}{l}$ , $\frac{\pi}{m}$ , and $\frac{\pi}{n}$ . The generators correspond to reflections in the sides of the triangle.

Arising from the geometrical nature of these groups,

$\frac{1}{l}+\frac{1}{m}+\frac{1}{n}=1$

is called the Euclidean case, because each group in this case represents symmetries of a certain tiling of the Euclidean plane by ordinary triangles whose angles add up to $π$ .

The case

$\frac{1}{l}+\frac{1}{m}+\frac{1}{n}>1$

is called the spherical case, because each such group represents symmetries of a tiling of the sphere by spherical triangles or Schwarz triangles, whose angles add up to a number greater than $π$ .

The case

$\frac{1}{l}+\frac{1}{m}+\frac{1}{n}<1$

is called the hyperbolic case, because each such group represents symmetries of a tiling of the hyperbolic plane by triangles whose angles add up to a number less than $π$ .

Specifically, the cases where $(l, m, n) = (2,2, n)$ , $(2,3,3)$ , $(2,3,4)$ , or $(2,3,5)$ represent tilings of the sphere; these are the only triangle groups which are finite. They correspond to the platonic solids: (2,2,n) to the family of degenerate solids formed by two identical regular polygons stuck together, (2,3,3) to the tetrahedron, (2,3,4) to the cube/octahedron dual, and (2,3,5) to the dodecahedron/icosahedron dual.

The cases $(l, m, n) = (2,3,6)$ , $(2,4,4)$ or $(3,3,3)$ represent tilings of the Euclidean plane; these are three special cases of wallpaper groups. All other cases, such as $(l, m, n) = (2,3,7)$ , represent tilings of the hyperbolic plane.

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

The Schwarz triangle map is a map of triangles to the upper half-plane.
The wallpaper group describes the tiling of the Euclidean plane.
Triangle groups are special cases of Coxeter groups.

[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.

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