Evaluating the exact infinitesimal values of area of Sierpinski's carpet and volume of Menger's sponge

TL;DR

This paper demonstrates how infinite and infinitesimal numbers enable exact calculation of the areas of Sierpinski's carpet and volumes of Menger's sponge at infinity, surpassing traditional limit-based methods.

Contribution

It introduces a novel approach using infinite and infinitesimal numbers to compute exact measures of fractals at infinity, improving upon traditional limit-based techniques.

Findings

Exact infinitesimal values of Sierpinski's carpet area calculated.

Exact infinitesimal volumes of Menger's sponge determined.

Traditional results are reproducible as finite approximations.

Abstract

Very often traditional approaches studying dynamics of self-similarity processes are not able to give their quantitative characteristics at infinity and, as a consequence, use limits to overcome this difficulty. For example, it is well know that the limit area of Sierpinski's carpet and volume of Menger's sponge are equal to zero. It is shown in this paper that recently introduced infinite and infinitesimal numbers allow us to use exact expressions instead of limits and to calculate exact infinitesimal values of areas and volumes at various points at infinity even if the chosen moment of the observation is infinitely faraway on the time axis from the starting point. It is interesting that traditional results that can be obtained without the usage of infinite and infinitesimal numbers can be produced just as finite approximations of the new ones.