  The process of alternately row scaling and column scaling a positive $n
\times n$ matrix $A$ converges to a doubly stochastic positive $n \times n$
matrix $S(A)$, called the \emph{Sinkhorn limit} of $A$. Exact formulae for the
Sinkhorn limits of certain symmetric positive $3\times 3$ matrices are
computed, and related problems in diophantine approximation are considered.
