New relatives of the Sierpinski gasket

TL;DR

This paper introduces new self-similar fractal sets related to the Sierpinski gasket, exhibiting dense regions and complex structures, expanding the understanding and modeling potential of fractals with separation conditions.

Contribution

It demonstrates how slight modifications to the Sierpinski gasket can produce self-similar sets with dense parts, revealing richer structures and classification challenges.

Findings

Self-similar sets with dense parts are possible under the open set condition.

The structure of the open set influences fractal density and complexity.

Computer searches can identify and classify these complex fractals.

Abstract

By slight modification of the data of the Sierpinski gasket, keeping the open set condition fulfilled, we obtain self-similar sets with very dense parts, similar to fractals in nature and in random models. This is caused by a complicated structure of the open set and is revealed only under magnification. Thus the family of self-similar sets with separation condition is much richer and has higher modelling potential than usually expected. An interactive computer search for such examples and new properties for their classification are discussed.