Triangle group

In mathematics, a triangle group is a group that can be realized geometrically by sequences of reflections across the sides of a triangle. The triangle can be an ordinary Euclidean triangle, a triangle on the sphere, or a hyperbolic triangle. Each triangle group is the symmetry group of a tiling of the Euclidean plane, the sphere or the hyperbolic plane by congruent triangles.

Definition

A triangle group $Delta(l,m,n)$ is a group of motions of the Euclidean plane, the two-dimensional sphere, or the hyperbolic plane. $Delta(l,m,n)$ is the group generated by the reflections in the sides of a triangle with angles $frac\{pi\}\{l\}$, $frac\{pi\}\{m\}$, and $frac\{pi\}\{n\}$(measured in radians). The product of the reflections in two adjacent sides is a rotation by the angle which is twice the angle between those 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

:$Delta(l,m,n)=langle\; a,b,c\; mid\; a^2,b^2,c^2,(ab)^l,(bc)^n,(ca)^m\; angle$where $l,m,n$ are integers greater than or equal to 2. Triangle groups are Coxeter groups with three generators.

Classification

Given any natural numbers $l,\; m,\; ngeq\; 2,$ exactly one of the classical two-dimensional geometries (Euclidean, spherical, or hyperbolic) admits a triangle with the angles $frac\{pi\}\{l\}$, $frac\{pi\}\{m\}$, and $frac\{pi\}\{n\}.$ 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.

In terms of the numbers $l,\; m,\; ngeq\; 2,$ 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 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 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 a unit 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"), "n"≥2. Spherical triangle groups can be identified with the symmetry groups of regular polyhedra in the three-dimensional Euclidean space: $Delta(2,3,3)$ corresponds to the tetrahedron, $Delta(2,3,4)$ to both the cube and the octahedron (which have the same symmetry group), $Delta(2,3,5)$ to both the dodecahedron and the icosahedron. The groups $Delta(2,2,n),\; ngeq\; 2$ can be interpreted as the symmetry groups of a family of "degenerate solids" formed by two identical regular "n"-gons joined together.

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.

The hyperbolic case

:

The triangle group is the infinite symmetry group of a 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.

von Dyck groups

Denote by $D(l,m,n)$ the subgroup of index 2 in $Delta(l,m,n)$, corresponding to the elements of the group that preserve the 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::$D(l,m,n)=langle\; x,y\; mid\; x^l,y^m,(xy)^n\; angle.$ 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 $l,m,n$.

ee also

* The Schwarz triangle map is a map of triangles to the upper half-plane.

References

* Robert Dawson [http://cs.smu.ca/faculty/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.)"