The Geometry of the Narayana Fractal

This paper examines the fractal nature of the Narayana fractal, an object defined by N = {(i, j) ∈ N× N : N(i + j + 1, j + 1) = 1 (mod 2)}. where N(n, k) = 1 n ( n k )( n k − 1 ) are the Narayana numbers. This object closely resembles a fractal derived from Pascal’s triangle. This similarity is used to prove that the Hausdorff dimension of the Narayana fractal is log 3/ log 2, and the limit of the Narayana fractal converges to the union of Sierpinski’s gasket with one additional point.