= Spherical triangle reflection group
{title2=$|G|=\frac4{1/p+1/q+1/r-1}$}
Reflections in the sides of a spherical triangle of angles $\pi/p,\pi/q,\pi/r$ generate a finite <group> when its images tessellate the sphere. The triangle area is $\pi(1/p+1/q+1/r-1)$ by the <spherical excess formula>. Label its three vertex types and continue those labels across each reflected side; they are consistent around every vertex and, because the sphere is <simply connected>, globally. The generated group preserves these labels, so a chamber stabilizer fixes all three vertices and is trivial. Dividing sphere area by chamber area gives its order. Up to permutation the possibilities are $(2,2,n)$ for $n\geq2$, and $(2,3,3),(2,3,4),(2,3,5)$, of orders $4n,24,48,120$.
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