The Shape of Existence: Tetrahedral Zinc and the Gap That Names Its Own Element

Biological zinc adopts tetrahedral coordination — 4-coordinate, four ligands, confirmed by over 4,000 structures in the Protein Data Bank. The tetrahedron is the minimum solid: the fewest faces that can enclose a volume. This paper identifies three structural properties of the tetrahedron that connect biological zinc to the eigenvalue tower of the companion papers. First, the interior measurement: an observer at the centroid sees three distances whose ratios trace a logarithmic spiral with growth factor √3, and whose outer-to-inner ratio R/r = 3 is the unique integer among all Platonic solids. Second, the gap: the inscribed sphere fills only π√3/18 ≈ 30.23% of the tetrahedral volume — the largest observation…