  We introduce cubic coordinates, which are integer words encoding intervals in
the Tamari lattices. Cubic coordinates are in bijection with interval-posets,
themselves known to be in bijection with Tamari intervals. We show that in each
degree the set of cubic coordinates forms a lattice, isomorphic to the lattice
of Tamari intervals. Geometric realizations are naturally obtained by placing
cubic coordinates in space, highlighting some of their properties. We consider
the cellular structure of these realizations. Finally, we show that the poset
of cubic coordinates is shellable.
