  In 1958, S. \'Swierczkowski proved that there cannot be a closed loop of
congruent interior-disjoint regular tetrahedra that meet face-to-face. Such
closed loops do exist for the other four regular polyhedra. It has been
conjectured that, for any positive \epsilon, there is a tetrahedral loop such
that its difference from a closed loop is less than \epsilon. We prove this
conjecture by presenting a very simple pattern that can generate loops of
tetrahedra in a rhomboid shape having arbitrarily small gap. Moreover,
computations provide explicit examples where the error is under $10^{-100}$ or
$10^{-10^{6}}$. The explicit examples arise from a certain Diophantine relation
whose solutions can be found through continued fractions; for more complicated
patterns a lattice reduction technique is needed.
