  We introduce a partial order on the set of all normal polytopes in R^d. This
poset NPol(d) is a natural discrete counterpart of the continuum of convex
compact sets in R^d, ordered by inclusion, and exhibits a remarkably rich
combinatorial structure. We derive various arithmetic bounds on elementary
relations in NPol(d), called "quantum jumps". The existence of extremal objects
in NPol(d) is a challenge of number theoretical flavor, leading to interesting
classes of normal polytopes: minimal, maximal, spherical. Minimal elements in
NPol(5) have played a critical role in disproving various covering conjectures
for normal polytopes in the 1990s. Here we report on the first examples of
maximal elements in NPol(4) and NPol(5), found by a combination of the
developed theory, random generation, and extensive computer search.
