Fractal dimension and topological invariants as methods to quantify complexity in Yayoi Kusama's paintings

TL;DR

This paper introduces quantitative methods using fractal dimension and Betti numbers to objectively measure complexity in abstract art, validated on synthetic images and applied to Yayoi Kusama's paintings, revealing nuanced insights into their structural complexity.

Contribution

It applies and validates fractal dimension and topological invariants as novel tools for analyzing artistic complexity, providing a more objective assessment method.

Findings

Fractal dimension of Kusama's works is similar to Pollock's drip paintings.

Betti numbers indicate disconnectedness rather than high complexity.

Results align with visual assessments of the artworks.

Abstract

Intricate patterns in abstract art many times can be wrongly characterized as being complex. Complexity can be an indicator of the internal dynamic of the whole system, regardless of the type of system in question, including art creation. In this investigation, we use two different techniques to objectively quantify complexity in abstract images: the fractal dimension and the value of the Betti numbers. We first validate our technique by considering synthetic images with a random distribution of dots, to then apply it to a series of `Net obsession' paintings by Yayoi Kusama. Surprisingly, we found that although the fractal dimension of her works in this series is comparable to those by Jackson Pollock in his dripping period, which could indicate a high level of complexity, the value of the Betti numbers do show disconnectedness and not high complexity. This is intuitively in agreement with the visual assessment of such works.