  We study the spectrum of the Fibonacci Hamiltonian and prove upper and lower
bounds for its fractal dimension in the large coupling regime. These bounds
show that as $\lambda \to \infty$, $\dim (\sigma(H_\lambda)) \cdot \log
\lambda$ converges to an explicit constant ($\approx 0.88137$). We also discuss
consequences of these results for the rate of propagation of a wavepacket that
evolves according to Schr\"odinger dynamics generated by the Fibonacci
Hamiltonian.
