A curious arithmetic of fractal dimension for polyadic Cantor sets

TL;DR

This paper introduces a novel arithmetic framework for the fractal dimension of polyadic Cantor sets, enabling calculus-like manipulation of their dimensions through well-defined operators.

Contribution

It presents a new arithmetic system for fractal dimensions, allowing continuous and differentiable operations on Cantor set dimensions based on their construction parameters.

Findings

Operators satisfy usual arithmetic properties

Enables calculus-based manipulation of fractal dimensions

Facilitates continuous and differentiable dimension analysis

Abstract

Fractal sets, by definition, are non-differentiable, however their dimension can be continuous, differentiable, and arithmetically manipulable as function of their construction parameters. A new arithmetic for fractal dimension of polyadic Cantor sets is introduced by means of properly defining operators for the addition, subtraction, multiplication, and division. The new operators have the usual properties of the corresponding operations with real numbers. The combination of an infinitesimal change of fractal dimension with these arithmetic operators allows the manipulation of fractal dimension with the tools of calculus.