Crystallization of space: Space-time fractals from fractal arithmetic

TL;DR

This paper explores the mathematical framework of space-time fractals using fractal arithmetic, enabling the formulation of physics directly on fractal sets, with explicit examples demonstrating the approach.

Contribution

It introduces a novel method to define calculus and algebra on fractals, allowing classical and quantum physics to be formulated within fractal space-time structures.

Findings

Fractals can be equipped with intrinsic arithmetic operations.

Calculus and algebra can be developed on fractals.

Explicit examples of fractal space-time constructions are provided.

Abstract

Fractals such as the Cantor set can be equipped with intrinsic arithmetic operations (addition, subtraction, multiplication, division) that map the fractal into itself. The arithmetics allows one to define calculus and algebra intrinsic to the fractal in question, and one can formulate classical and quantum physics within the fractal set. In particular, fractals in space-time can be generated by means of homogeneous spaces associated with appropriate Lie groups. The construction is illustrated by explicit examples.