  In this paper, we prove that given any \Pi^0_1 subset $P$ of $\{0,1\}^\NN$
there is a tileset $\tau$ with a set of configurations $C$ such that
$P\times\ZZ^2$ is recursively homeomorphic to $C\setminus U$ where $U$ is a
computable set of configurations. As a consequence, if $P$ is countable, this
tileset has the exact same set of Turing degrees.
