  A point in the $d$-dimensional integer lattice $\mathbb{Z}^d$ is primitive
when its coordinates are relatively prime. Two primitive points are multiples
of one another when they are opposite, and for this reason, we consider half of
the primitive points within the lattice, the ones whose first non-zero
coordinate is positive. We solve the packing problem that asks for the largest
possible number of such points whose absolute values of any given coordinate
sum to at most a fixed integer $k$. We present several consequences of this
result at the intersection of geometry, number theory, and combinatorics. In
particular, we obtain an explicit expression for the largest possible diameter
of a lattice zonotope contained in the hypercube $[0,k]^d$ and, conjecturally
of any lattice polytope in that hypercube.
