Irregularity Index and Spherical Densities of the Penta-Sierpinski Gasket

Abstract

We compute the centred Hausdorff measure, C s (P) ∼ 2.44, and the packing measure, P s (P) ∼ 6.77, of the penta-Sierpinski gasket, P, with explicit error bounds. We also compute the full spectra of asymptotic spherical densities of these measures in P, which, in contrast with that of the Sierpinski gasket, consists of a unique interval. These results allow us to compute the irregularity index of P, I(P) ∼ 0.6398, which we define for any self-similar set E with open set condition as I(E) = 1 − C s (E) P s (E) .