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 will see 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 will present the cellular structure of these realizations. Finally, we will see a proof generalizing the result of Björner and Wachs about the EL-shellability of Tamari posets for Tamari interval posets.