A Prime-Indexed Logarithmic Spiral Geometric Graph via the Golden Angle

We introduce the Prime Spiral Geometric Graph (PSGG), a geometric construction in which each prime pi generates a regular pi-gon whose center lies on a logarithmic spiral with angular step equal to the Golden Angle ω ≈ 137.508◦ . The full vertex set for the first N = 137 primes has cardinality |V | = 48,494; endowing it with a k-nearest-neighbour edge set yields a dense geometric graph G = (V, E). We prove that the number of vertices inside a disk of radius R satisfies |V (R)| ∼ k 2 2 log2φ (log R) 2 log log R (R → ∞), a consequence of the classical prime-sum asymptotics of Landau (1909) and Massias–Robin (1996). Computationally, G splits into 129 components for k = 6 nearest neighbours and becomes connected for k ≥ 10. We place the construction in the context of prior work (Ulam spiral, Sacks spiral, phyllotaxis models, polygonal spiral sequences), identify the dual role of each prime as both spiral index and polygon vertex count as the novel element, and list seven open problems connecting the PSGG to spectral graph theory, algebraic topology, and the distribution of prime gap