Fractal surfaces from simple arithmetic operations

TL;DR

This paper demonstrates that simple bitwise operations on real numbers can generate fractal surfaces with varying roughness, providing a general theoretical framework for understanding their formation and properties.

Contribution

It introduces a comprehensive theory for deterministic bitwise operations on finite alphabets, linking these operations to the emergence of fractal surfaces and their roughness characteristics.

Findings

Bitwise operations produce fractal surfaces with scale-invariant roughness.

The roughness exponent $H$ determines the coarseness of the generated surfaces.

Models show a direct relationship between bitwise operations and surface complexity.

Abstract

Fractal surfaces ('patchwork quilts') are shown to arise under most general circumstances involving simple bitwise operations between real numbers. A theory is presented for all deterministic bitwise operations on a finite alphabet. It is shown that these models give rise to a roughness exponent $H$ that shapes the resulting spatial patterns, larger values of the exponent leading to coarser surfaces.