  In the context of metric structures introduced by Ben Yaacov, Berenstein,
Henson, and Usvyatsov, we exhibit an explicit encoding of metric structures in
countable signatures as pure metric spaces in the empty signature, showing that
such structures are universal for bi-interpretation among metric structures
with positive diameter. This is analogous to the classical encoding of
arbitrary discrete structures in finite signatures as graphs, but is stronger
in certain ways and weaker in others. There are also certain fine grained
topological concerns with no analog in the discrete setting.
