caa a22 4500 44588455X CHVBK 20180317145555.0 cr unu---uuuuu 170323e20110201xx s 000 0 eng 10.1007/s10711-010-9502-y doi (NATIONALLICENCE)springer-10.1007/s10711-010-9502-y Similar dissection of sets [Elektronische Daten] [Shigeki Akiyama, Jun Luo, Ryotaro Okazaki, Wolfgang Steiner, Jörg Thuswaldner] 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 $${D = X \cup f_1(X) \cup \cdots \cup f_k(X)}$$ is satisfied with a non-overlapping union? We will prove that such a set X exists for certain choices of D and {f 1, . . . , f k}. The solutions X will 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 $${a \ge (3+\sqrt{5})/2}$$. Springer Science+Business Media B.V., 2010 Iterated function system nationallicence Dissection nationallicence Tiling nationallicence Akiyama Shigeki Department of Mathematics, Faculty of Science, Niigata University, Ikarashi 2-8050, 950-2181, Niigata, Japan aut Luo Jun School of Mathematics and Computational Science, Sun Yat-Sen University, 510275, Guangzhou, China aut Okazaki Ryotaro Department of Knowledge Engineering and Computer Sciences, Doshisha University, 1-3 Tataramiyakodani, Kyotanabe-shi, 610-0394, Kyoto-fu, Japan aut Steiner Wolfgang LIAFA, CNRS UMR 7089, Université Paris Diderot - Paris 7, Case 7014, 75205, Paris Cedex 13, France aut Thuswaldner Jörg Department of Mathematics and Statistics, University of Leoben, Franz-Josef-Strasse 18, 8700, Leoben, Austria aut Geometriae Dedicata Springer Netherlands 150/1(2011-02-01), 233-247 0046-5755 150:1<233 2011 150 10711 https://doi.org/10.1007/s10711-010-9502-y text/html Onlinezugriff via DOI 1 research-article jats NATIONALLICENCE 856 40 https://doi.org/10.1007/s10711-010-9502-y text/html Onlinezugriff via DOI NATIONALLICENCE 700 1- Akiyama Shigeki Department of Mathematics, Faculty of Science, Niigata University, Ikarashi 2-8050, 950-2181, Niigata, Japan aut NATIONALLICENCE 700 1- Luo Jun School of Mathematics and Computational Science, Sun Yat-Sen University, 510275, Guangzhou, China aut NATIONALLICENCE 700 1- Okazaki Ryotaro Department of Knowledge Engineering and Computer Sciences, Doshisha University, 1-3 Tataramiyakodani, Kyotanabe-shi, 610-0394, Kyoto-fu, Japan aut NATIONALLICENCE 700 1- Steiner Wolfgang LIAFA, CNRS UMR 7089, Université Paris Diderot - Paris 7, Case 7014, 75205, Paris Cedex 13, France aut NATIONALLICENCE 700 1- Thuswaldner Jörg Department of Mathematics and Statistics, University of Leoben, Franz-Josef-Strasse 18, 8700, Leoben, Austria aut NATIONALLICENCE 773 0- Geometriae Dedicata Springer Netherlands 150/1(2011-02-01), 233-247 0046-5755 150:1<233 2011 150 10711 Metadata rights reserved Springer special CC-BY-NC licence nationallicence BK010053 XK010053 XK010000 NATIONALLICENCE NATIONALLICENCE NL-springer