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Triangle group

From Biocrawler, the free encyclopedia.

In mathematics, the triangle groups are finite groups. Some, but not all, of these groups represent the tiling of surfaces by triangles.

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Definition

A triangle group is defined by the group presentation

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

where $l, m, n$ are positive integers.

A group with this presentation corresponds to a triangle; roughly, the generators are reflections in its sides and its angles are $π / l,π / m,π / n$ .

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 refered to as von Dyck groups.

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

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

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 $l, m, n$ .

Arising from the geometrical nature of these groups,

1 / l + 1 / m + 1 / n > 1

is called the spherical case; some of these groups tile the sphere with Schwarz triangles. The case

1 / l + 1 / m + 1 / n = 1

is called the Euclidean case, some of these groups tile the Euclidean plane by ordinary triangles.

1 / l + 1 / m + 1 / n < 1

is called the hyperbolic case, some of these groups tile the hyperbolic plane by hyperbolic triangles.

Specifically, (2,2,n), (2,3,3), (2,3,4) and (2,3,5) tile the sphere. The plane is tiled by (2,3,6) and the hyperbolic plane by (2,3,7).

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References

Robert Dawson Some spherical tilings (http://cs.smu.ca/faculty/dawson/images4.html) (undated, earlier than 2004) (Shows a number of interesting sphere tilings, most of which are not triangle group tilings.)

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