HEXAGONAL METRICS AND GOLDEN-RATIO SPIRALS: A METRIC ORIGIN FOR GOLDEN-ANGLE PHYLLOTAXIS

Hexagonal order recurs throughout nature as a locally energy-minimising arrangement for equal-radius interactions, from soap froths to bee combs to the densest circle packings. Logarithmic spirals, and in particular the golden-angle phyllotactic spiral, also arise across biological and physical systems, where new units accrete by a constant turn at multiplicatively increasing radii. This paper advances a coherent mathematical account linking these two motifs by proposing that the metric governing local interactions plays the decisive role. We formalise a hexagonal metric as a Minkowski norm whose unit ball is a regular hexagon and analyse constant-turn, multiplicative growth in this metric. We prove that such growth generates logarithmic spirals in the continuum limit and that the choice of the turn angle, maximising the asymptotic uniformity of placements, is uniquely achieved by the golden angle α* = 2π/φ², where φ = (1+√5)/2 is the golden ratio. The optimality criterion is expressed through minimal pairwise distance under the hexagonal norm, which we connect to the Diophantine properties of the turn angle. Specifically, we show that the golden ratio's extremal irrationality allows the spiral to best avoid the "resonant" directions of the sixfold anisotropic potential, thereby minimizing crowding. The theory explains why systems that experience local interactions with sixfold symmetry naturally express golden-angle spirals at mesoscopic scales, even when no overt hexagonal lattice is visible in Euclidean space. This yields testable predictions distinguishing this metric origin from purely Euclidean models, including specific parastichy counts, anisotropic Voronoi cell statistics, and the presence of spectral sidebands at multiples of π/3 in the angular structure factor.