THE BOX-COUNTING DIMENSION OF PASCAL’S TRIANGLE $rmodp$

We consider Pascal’s Triangle $rmodp$ to be the entries of Pascal’s Triangle that are congruent to $rmodp$. Such a representation of Pascal’s Triangle exhibits fractal-like structures. When the Triangle is mapped to a subset of the unit square, we show that such a set is nonempty and exists as a limit of a sequence of coarse approximations. We then show that for any given prime $p$, any such sequence converges to the same set, regardless of the residue(s) considered. As an obvious consequence, this allows us to conclude that the fractal (box-counting) dimension of this nonempty, compact representation of Pascal’s Triangle $rmodp$ is independent of $r$.