Fractal mountains and binary random loops

TL;DR

This paper introduces a novel algebraic approach to constructing fractal mountain-like functions using binary representations of loops, enabling explicit calculation of their values and integrals despite their chaotic appearance.

Contribution

It presents a new algebraic method for generating fractal functions based on binary loop representations, differing from traditional geometric approaches.

Findings

Functions exhibit chaotic and fractal forms resembling mountains.

Values and integrals of these functions can be explicitly computed.

The approach bridges algebraic structures with fractal geometry.

Abstract

We discuss a variation of Takagi curves based, however, more on algebraic than geometric principles. Namely, we construct functions of loops in a special binary representation. The graph of these functions usually has chaotic and fractal forms, sometimes recall mountain landscapes. Nevertheless, the values in a dense set of points and even the integral of these functions can be calculated explicitly.