 Triangle group

A triangle group is generated by reflections in the sides of a triangular fundamental domain in spherical geometry, Euclidean geometry or hyperbolic geometry. The angles $\pi /p,\pi /q,\pi /r$ determine the geometry through the sign of $1/p+1/q+1/r-1$. Its orientation-preserving subgroup has group presentation $⟨x,y,z\mid x^p=y^q=z^r=xyz=1⟩$.

In mathematics, particularly in the fields of group theory and geometric group theory, a **triangle group** is a type of group that can be defined geometrically by its presentation and associated with the symmetries of triangles in a certain way.