  We present the first algorithm to morph graphs on the torus. Given two
isotopic essentially 3-connected embeddings of the same graph on the Euclidean
flat torus, where the edges in both drawings are geodesics, our algorithm
computes a continuous deformation from one drawing to the other, such that all
edges are geodesics at all times. Previously even the existence of such a morph
was not known. Our algorithm runs in $O(n^{1+\omega/2})$ time, where $\omega$
is the matrix multiplication exponent, and the computed morph consists of
$O(n)$ parallel linear morphing steps. Existing techniques for morphing planar
straight-line graphs do not immediately generalize to graphs on the torus; in
particular, Cairns' original 1944 proof and its more recent improvements rely
on the fact that every planar graph contains a vertex of degree at most 5. Our
proof relies on a subtle geometric analysis of 6-regular triangulations of the
torus. We also make heavy use of a natural extension of Tutte's spring
embedding theorem to torus graphs.
