  We construct an embedding $\Phi$ of $[0,1]^{\infty}$ into $Ham(M, \omega)$,
the group of Hamiltonian diffeomorphisms of a suitable closed symplectic
manifold $(M, \omega)$. We then prove that $\Phi$ is in fact a quasi-isometry.
After imposing further assumptions on $(M, \omega)$, we adapt our methods to
construct a similar embedding of $\mathbb{R} \oplus [0,1]^{\infty}$ into either
$Ham(M, \omega)$ or $\widetilde{Ham}(M, \omega)$, the universal cover of
$Ham(M, \omega)$. Along the way, we prove results related to the filtered Floer
chain complexes of radially symmetric Hamiltonians. Our proofs rely heavily on
a continuity result for barcodes (as presented in the work of M. Usher and J.
Zhang) associated to filtered Floer homology viewed as a persistence module.
