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.

A fractafold, a space that is locally modeled on a specified fractal, is the
fractal equivalent of a manifold. For compact fractafolds based on the
Sierpinski gasket, it was shown by the first author how to compute the discrete
spectrum of the Laplacian in terms of the spectrum of a finite graph Laplacian.
A similar problem was solved by the second author for the case of infinite
blowups of a Sierpinski gasket, where spectrum is pure point of infinite
multiplicity. Both works used the method of spectral decimations to obtain
explicit description of the eigenvalues and eigenfunctions.