The Takagi Curve and the $\beta$-Cantor Function from Mechanical Laws

TL;DR

This paper demonstrates that mechanical laws can naturally produce fractals like the Takagi curve and $eta$-Cantor functions, revealing intrinsic links between physical processes and fractal mathematics.

Contribution

It shows how elasticity theory applied to branching structures naturally generates fractal patterns without algorithmic intervention.

Findings

Fractal tree crowns emerge from mechanical stress analysis.

Vertical displacements follow the Takagi curve.

Horizontal displacements relate to $eta$-Cantor functions.

Abstract

This work shows that fractals can be obtained from Mechanical Laws without being forced by any algorithm, closing the gap between the Platonic world of Mathematics and Nature. Fractal tree crown directly emerges when applying elasticity theory to branching stresses in a binary tree. Vertical displacements of nodes are given by the Takagi curve, while the horizontal ones are given by a linear combination of inverses of $\beta$-Cantor functions. In addition, both fractal dimensions are related, which suggests a deeper connection between the Takagi Curve and the $\beta$-Cantor function.