On the Centred Hausdorff Measure of the Sierpinski Gasket

TL;DR

This paper demonstrates that the centred Hausdorff measure of the Sierpinski gasket is both continuous- and algorithmically-computable, providing precise numerical estimates and bounds for this measure and related spherical Hausdorff measure.

Contribution

It introduces a method to compute the centred Hausdorff measure of the Sierpinski gasket and establishes its approximate value along with bounds for the spherical Hausdorff measure.

Findings

Centred Hausdorff measure is C- and A-computable.

Estimated value of the measure is approximately 1.0049.

Bound for the spherical Hausdorff measure is approximately 0.8616 to 0.8619.

Abstract

We show that the centred Hausdorff measure, $C^{s}(S),$ with $s=\frac{\log3}{\log2},$ of the Sierpinski gasket $S$, is $C$-computable (continuous-computable), in the sense that its value is the solution of the minimisation problem of a continuous function on a compact domain. We also show that $C^{s}(S)$ is $A$-computable (algorithmic-computable) in the sense that there is an algorithm that converges to $C^{s}(S),$ with error bounds tending to zero. Using this algorithm and bounds we show that $C^{s}(S)\sim1.0049,$ and we establish a conjecture for the value of the spherical Hausdorff $s$-measure of $S,$ $\mathcal{H}_{sph}^{s}(S)\sim0.8616$, and provide an upper bound for it, $\mathcal{H}_{sph}^{s}(S)\leq0.8619.$