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

$3$D Farey graph, lambda lengths and $SL_2$-tilings