Integer triangle formula_112 and formula_113 generate the same triple. Unfortunately the two pairs can both be of gcd=3, so we can't avoid duplicates by simply skipping that case. Instead, duplicates can be avoided by formula_27 going only till formula_115. Note that we still need to divide by 3 if gcd=3. The only solution for formula_116 under the above constraints is formula_117 for formula_118. With this additional formula_119 constraint all triples can be generated uniquely. An Eisenstein triple is a set of integers which are the lengths of the sides of a triangle where one of the angles is 60 degrees. Integer triangles
Triangle group plane, the two-dimensional sphere, the real projective plane, or the hyperbolic plane generated by the reflections in the sides of a triangle with angles π/"l", π/"m" and π/"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, 2π/"l", 2π/"m" and 2π/"n". 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: It is a theorem that all other relations between "a, b, c" are consequences