  We investigate the relation between simple random walks on repeated
barycentric subdivisions of a triangle and a self-similar fractal, Strichartz
hexacarpet, which we introduce. We explore a graph approximation to the
hexacarpet in order to establish a graph isomorphism between the hexacarpet
approximations and Barycentric subdivisions of the triangle, and discuss
various numerical calculations performed on the these graphs. We prove that
equilateral barycentric subdivisions converge to a self-similar geodesic metric
space of dimension log(6)/log(2), or about 2.58. Our numerical experiments give
evidence to a conjecture that the simple random walks on the equilateral
barycentric subdivisions converge to a continuous diffusion process on the
Strichartz hexacarpet corresponding to a different spectral dimension
(estimated numerically to be about 1.74).
