structural:scholar_edition
Generalization to other moduli

A generalization of the Sierpiński triangle can also be generated using Pascal's triangle if a different modulus P {\displaystyle P} is used. Iteration n {\displaystyle n} can be generated by taking a Pascal's triangle with P n {\displaystyle P^{n}} rows and coloring numbers by their value modulo P {\displaystyle P}. As n {\displaystyle n} approaches infinity, a fractal is generated.

The same fractal can be achieved by dividing a triangle into a tessellation of P 2 {\displaystyle P^{2}} similar triangles and removing the triangles that are upside-down from the original, then iterating this step with each smaller triangle.

Conversely, the fractal can also be generated by beginning with a triangle and duplicating it and arranging n (n + 1) 2 {\displaystyle {\tfrac {n(n+1)}{2}}} of the new figures in the same orientation into a larger similar triangle with the vertices of the previous figures touching, then iterating that step.