HAL : hal-00449889, version 1

arXiv : 1001.4203

DOI : 10.1007/s10711-010-9502-y

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Geometriae Dedicata 150, 1 (2011) 233-247

Similar dissection of sets

Shigeki Akiyama 1, Jun Luo 2, Ryotaro Okazaki 3, Wolfgang Steiner 4, Jörg Thuswaldner 5

(02/2011)

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 $f_1,\ldots,f_k$ be injective continuous mappings. Does there exist a set $X$ such that $D = X \cup f_1(X) \cup \ldots \cup f_k(X)$ is satisfied with a non-overlapping union? We prove that such a set $X$ exists for certain choices of $D$ and $\{f_1,\ldots,f_k\}$. 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 $a \ge (3+\sqrt{5})/2$.

1 :	Department of Mathematics
	Niigata University
2 :	School of Mathematics and Computational Science
	Sun Yat-sen University
3 :	Department of Knowledge Engineering and Computer Sciences
	Doshisha University
4 :	Laboratoire d'informatique Algorithmique : Fondements et Applications (LIAFA)
	CNRS : UMR7089 – Université Paris VII - Paris Diderot
5 :	Montan Universität Leoben
	Montanuniversität Leoben

Domaine

:

Mathématiques/Géométrie métrique

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Contributeur : Wolfgang Steiner <>

Soumis le : Samedi 23 Janvier 2010, 16:08:37

Dernière modification le : Vendredi 2 Décembre 2011, 00:37:18