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 \({D = X \cup f_1(X) \cup \cdots \cup f_k(X)}\) 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 \({a \ge (3+\sqrt{5})/2}\).
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This research was supported by the Japanese Ministry of Education, Culture, Sports, Science and Technology, Grant-in Aid for fundamental research 21540010, 2009–2011, by the project 10601069 of the National Natural Science Foundation of China, by the project ANR–06–JCJC–0073 “DyCoNum” of the French Agence Nationale de la Recherche, by the project S9610 of the Austrian Science Foundation, by the Amadée grant FR–13–2008 and the PHC Amadeus grant 17111UB.
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Akiyama, S., Luo, J., Okazaki, R. et al. Similar dissection of sets. Geom Dedicata 150, 233–247 (2011). https://doi.org/10.1007/s10711-010-9502-y
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DOI: https://doi.org/10.1007/s10711-010-9502-y