The spatial distribution of persistent spins at zero temperature in the pure two-dimensional Ising model is investigated numerically. A persistence correlation length, ξ(t) ∼ tZ, is identified such that for length scales r ≪ ξ(t) the persistent spins form a fractal with dimension df; for length scales r ≫ ξ(t) the distribution of persistent spins is homogeneous. The zero-temperature persistence exponent, θ, is found to satisfy the scaling relation θ = Z(2 - df) with θ = 0.209 ± 0.002 of Jain (Jain S 1999 Phys. Rev. E 59 R2493), Z = 1/2 and df ∼ 1.58.