The crossing number and the unit-distance crossing number of the Hamming graphs H(d,3)=K_3^{box d}, and their realizations over many coordinate fields

Abstract

The Hamming graph H(d,3)=K_3^{box d} (n=3^d vertices) is the graph of single-symbol errors of ternary codes, yet we ask a geometric question of it: can it be drawn in the plane with every edge exactly one unit long and no two non-adjacent vertices a unit apart? It can, for every d, joining two problems into one. Where does it live? As a Minkowski sum of unit triangles, each H(d,3) has a hidden flexibility carrying its coordinates up the constructibility ladder (compass, origami, and beyond), so one graph is realizable over many number fields at once, in the plane and in R^3 (edim(H(d,q))=q-1). Yet almost every faithful realization is transcendental: the Galois picture is a measure-zero shadow of a vast transcendental continuum, matching the exists-R hardness of unit-distance recognition. How crowded must a unit drawing be? We separate the ordinary crossing number from a unit-distance crossing number (absent from Schaefer's survey), and one recursion H(d,3)=H(d-1,3) box K_3 controls both. We prove cr(H)=Theta(n^2), with sharp one-page constant 7/6; and Omega(n^2) <= udcr(H) <= O(n^2 log n) by concentration, plus a closed-form majorant (3/2)n^2(L^2-L+1), L=log_3 n. The least-crossing drawing we find is constructible (origami): field and crossings are two readings of one geometry. All claims are verified computationally; the lower bound udcr=Omega(n^2 log n) is the central open problem.