Abstract

We propose a new geometric representation of divisibility based on binary matrices and the C4
symmetry group. The arithmetic properties of prime and composite numbers — well known in
themselves — acquire in this representation a visual manifestation that reveals non-obvious
aspects. The main results are: (1) the N-1 diagonal of the base matrix encodes the divisors of N (an
elementary consequence, but geometrically significant); (2) the diagonal density D% in the C4 overlay
converges to 1/3 (a new quantitative result); (3) the peripheral density satisfies P% ~ 6.17 tau(N)/N^2,
linking geometry to the divisor function (new); (4) visual observation reveals Crystalline Numbers —
composites whose collisions fall exclusively on the diagonal and border — and computational
verification on 1000 numbers shows that N is crystalline if and only if N+1 is prime (for N >= 16,
zero exceptions). The proof relies on an elementary property (two numbers with prime sum are
coprime), but its geometric manifestation is new and constitutes a geometric primality test.