  We explore a three-dimensional counterpart of the Farey tessellation and its
relations to Penner's lambda lengths and $SL_2$-tilings. In particular, we
prove a three-dimensional version of Ptolemy relation, and generalise results
of Ian Short to classify tame $SL_2$-tilings over Eisenstein integers in terms
of pairs of paths in the 3D Farey graph.
