A Fractal Comparison of Escher and Koch Tessellations

TL;DR

This paper compares M.C. Escher's hyperbolic tessellations with fractal patterns like the Koch Snowflake and Pollock's bifractals, revealing both differences and surprising similarities through scaling analysis.

Contribution

It introduces a quantitative box counting analysis to distinguish Escher's hyperbolic patterns from fractals and uncovers unexpected bifractal-like features in Escher's work.

Findings

Escher's patterns follow hyperbolic scaling laws.

Distinct visual differences between hyperbolic and monofractal patterns.

Unexpected bifractal similarities between Escher's tessellations and Pollock's paintings.

Abstract

M.C. Eschers tessellations have captured the imaginations of both artists and mathematicians. Circle Limit III is the most intricate of his tessellations, featuring patterns that repeat at increasingly fine scales. Although his patterns follow a scaling law determined by hyperbolic geometry, his work is often mistakenly described as following fractal geometry. Here, we perform a 'box counting' scaling analysis on Circle Limit III and an equivalent monofractal pattern based on a Koch Snowflake. Whereas our analysis highlights the expected visual differences between Eschers hyperbolic patterns and the simple monofractal, the analysis also identifies unexpected similarities between Eschers work and the bifractal poured paintings of Jackson Pollock.