The first explicit realization of the conjecture that phason dynamics leads
to self-diffusion in quasicrystals is presented for the icosahedral Ammann
tilings. On short time scales, the transport is found to be subdiffusive with
the exponent $\beta\approx0.57(1)$, while on long time scales it is consistent
with normal diffusion that is up to an order of magnitude larger than in the
typical room temperature vacancy-assisted self-diffusion. No simple finite-size
scaling is found, suggesting anomalous corrections to normal diffusion, or
existence of at least two independent length scales.