  We give the quasi--Euclidean classification of the maximal (with respect to
the $f$--vector) alcoved polyhedra. The $f$--vector of these maximal convex
bodies is $(20,30,12)$, so they are simple dodecahedra. We find eight
quasi--Euclidean classes. This classification, which preserves angles, is finer
than the known combinatorial classification (found in 2012 by Jim\'{e}nez and
de la Puente), which has only six classes. Each alcoved polyhedron
$\mathcal{P}$ is represented by a unique visualized idempotent matrix $A$. Some
2--minors of $A$ are invariants of $\mathcal{P}$: they are the tropical
edge--lengths of $\mathcal{P}$.
