Consider a planar Brownian motion run for finite time. The frontier or
``outer boundary'' of the path is the boundary of the unbounded component of
the complement. Burdzy (1989) showed that the frontier has infinite length. We
improve this by showing that the Hausdorff dimension of the frontier is
strictly greater than 1. (It has been conjectured that the Brownian frontier
has dimension 4/3, but this is still open.) The proof uses Jones's Traveling
Salesman Theorem and a self-similar tiling of the plane by fractal tiles known
as Gosper Islands