Central Open Sets Tilings

TL;DR

This paper presents a novel method for constructing tilings of  spaces using iterated function systems, focusing on central open sets and their properties, with practical computation techniques and illustrative examples.

Contribution

It introduces a new approach to generate tilings based on central open sets of iterated function systems, expanding understanding and visualization of fractal tilings.

Findings

Method for constructing tilings using iterated function systems and cost functions

Practical techniques for visualizing central open sets

Examples demonstrating properties of the tilings

Abstract

We introduce a method for constructing collections of subsets of $\mathbb{R}^{n}$, using an iterated function system, a set $T,$ and a cost function. We refer to these collections as tilings. The special case where $T$ is the central open set of an iterated function system that obeys the open set condition is emphasized. The notion of the central open set associated with an iterated function system of similitudes, introduced in 2005 by Bandt, Hung, and Rao, is reviewed. A practical method for calculating pictures of central open sets is described. Some general properties and examples of the tilings are presented.