Homogeneous 3-dimensional permutation structures

We provide a classification of the homogeneous 3-dimensional permutation structures, i.e. homogeneous structures in a language of 3 linear orders, partially answering a 2002 question of Cameron. We also arrive at a natural description of all known homogeneous finite-dimensional permutation structures by modifying the language used in the construction from [Samuel Braunfeld, Electronic Journal of Combinatorics, 2016], completing the catalog begun there.