Similar dissection of sets

TL;DR

This paper generalizes questions about dissecting geometric shapes into similar parts, providing solutions for certain sets and functions, and applying the theory to dissect an equilateral triangle into three similar parts with specific area ratios.

Contribution

It extends Gardner's classical dissection questions to a broader context, offering existence results for non-overlapping unions of sets under iterated function systems.

Findings

Existence of sets X satisfying dissection conditions for certain D and functions

Dissection of an equilateral triangle into three similar parts with area ratio ≥ (3+√5)/2

Use of attractors of iterated function systems with condensation in solutions

Abstract

In 1994, Martin Gardner stated a set of questions concerning the dissection of a square or an equilateral triangle in three similar parts. Meanwhile, Gardner's questions have been generalized and some of them are already solved. In the present paper, we solve more of his questions and treat them in a much more general context. Let $D\subset \mathbb{R}^d$ be a given set and let $f_1,...,f_k$ be injective continuous mappings. Does there exist a set $X$ such that $D = X \cup f_1(X) \cup ... \cup f_k(X)$ is satisfied with a non-overlapping union? We prove that such a set $X$ exists for certain choices of $D$ and $\{f_1,...,f_k\}$. The solutions $X$ often turn out to be attractors of iterated function systems with condensation in the sense of Barnsley. Coming back to Gardner's setting, we use our theory to prove that an equilateral triangle can be dissected in three similar copies whose areas have ratio $1:1:a$ for $a \ge (3+\sqrt{5})/2$.