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
Sierpinski triangle converge on the Sierpinski triangle. This is what is happening with the triangle above, but any other set would suffice. If one takes a point and applies each of the transformations "d", "d", and "d" to it randomly, the resulting points will be dense in the Sierpinski triangle, so the following algorithm will again generate arbitrarily close approximations to it: Start by labeling p, p and p as the corners of the Sierpinski triangle, and a random point v. Set , where "r" is a random number 1, 2 or 3. Draw the points v to v. If the first