On certain families of planar patterns and fractals

TL;DR

This survey explores various families of planar fractals, especially Sierpiński carpets, highlighting their topological and geometrical properties through a unified framework of patterns.

Contribution

It provides a comprehensive overview and unifying approach to recent developments in the study of Sierpiński carpet families and their properties.

Findings

Unified framework for analyzing planar fractals.

Topological properties of limit net sets and labyrinth fractals.

Geometrical characteristics of generalized Sierpiński carpets.

Abstract

This survey article is dedicated to some families of fractals that were introduced and studied during the last decade, more precisely, families of Sierpi\'nski carpets: limit net sets, generalised Sierpi\'nski carpets and labyrinth fractals. We give a unifying approach of these fractals and several of their topological and geometrical properties, by using the framework of planar patterns.