Triangle Groups Δ(l,m,n)

Explore tilings of the sphere, Euclidean plane, and hyperbolic plane generated by reflections in the sides of a triangle with angles π/l, π/m, π/n.

← Plane Symmetries
Presentation:
-
Geometry: -
Curvature (k): -

The geometry is determined by the angle excess of the fundamental triangle. By Gauss–Bonnet, for a geodesic triangle with angles α, β, γ:

α + β + γ − π = K · Area

Here α = π/l, β = π/m, γ = π/n, so the sign of

1/l + 1/m + 1/n − 1

determines K: positive → spherical, zero → Euclidean, negative → hyperbolic.