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
Schwarz triangle function a vertex of the polygon "P" makes an angle of less than 2/3 with each of the edges of the polygon at that vertex if exactly two triangles of "P" meet at the vertex, since each has an angle less than or equal to /3 at that vertex. To check this is true when three triangles of "P" meet at the vertex, "C" say, suppose that the middle triangle has its base on a side "AB" of "P". By induction the radii "OA" and "OB" makes angles of less than or equal to 2/3 with the edge "AB". In this