Geometriae Dedicata

, Volume 150, Issue 1, pp 233–247 | Cite as

Similar dissection of sets

  • Shigeki Akiyama
  • Jun Luo
  • Ryotaro Okazaki
  • Wolfgang Steiner
  • Jörg Thuswaldner
Original Paper

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 \({a \ge (3+\sqrt{5})/2}\).

Keywords

Iterated function system Dissection Tiling 

Mathematics Subject Classification (2000)

28A80 52C20 52C21 

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References

  1. 1.
    Barnsley M.F.: Fractals Everywhere, 2nd edn. Academic Press Professional, Boston, MA (1993). Revised with the assistance of and with a foreword by Hawley Rising, IIIGoogle Scholar
  2. 2.
    Broomhead D., Montaldi J., Sidorov N.: Golden gaskets: variations on the Sierpiński sieve. Nonlinearity 17, 1455–1480 (2004)MathSciNetMATHCrossRefGoogle Scholar
  3. 3.
    Chun B.-K., Liu A., van Vliet D.: Dissecting squares into similar rectangles. Crux Math. 22, 241–248 (1996)Google Scholar
  4. 4.
    Gardner M.: Six challenging dissection tasks. Quantum 4, 26–27 (1994)Google Scholar
  5. 5.
    Gardner, M.: A Gardner’s Workout. A K Peters Ltd., Natick, MA, Training the mind and entertaining the spirit (2001)Google Scholar
  6. 6.
    Hatcher A.: Algebraic Topology. Cambridge University Press, Cambridge (2002)MATHGoogle Scholar
  7. 7.
    Hutchinson J.E.: Fractals and self-similarity. Indiana Univ. Math. J. 30, 713–747 (1981)MathSciNetMATHCrossRefGoogle Scholar
  8. 8.
    Liping Y., Yuqin Z., Ren D.: Trisecting a parallelogram. Appl. Math. J. Chin. Univ. Ser. B 17, 307–312 (2002)MATHCrossRefGoogle Scholar
  9. 9.
    Maltby S.J.: Trisecting a rectangle. J. Comb. Theory Ser. A 66, 40–52 (1994)MathSciNetMATHCrossRefGoogle Scholar
  10. 10.
    Stewart I.N., Wormstein A.: Polyominoes of order 3 do not exist. J. Comb. Theory Ser. A 61, 130–136 (1992)MathSciNetMATHCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media B.V. 2010

Authors and Affiliations

  • Shigeki Akiyama
    • 1
  • Jun Luo
    • 2
  • Ryotaro Okazaki
    • 3
  • Wolfgang Steiner
    • 4
  • Jörg Thuswaldner
    • 5
  1. 1.Department of Mathematics, Faculty of ScienceNiigata UniversityNiigataJapan
  2. 2.School of Mathematics and Computational ScienceSun Yat-Sen UniversityGuangzhouChina
  3. 3.Department of Knowledge Engineering and Computer SciencesDoshisha UniversityKyoto-fuJapan
  4. 4.LIAFA, CNRS UMR 7089Université Paris Diderot - Paris 7Paris Cedex 13France
  5. 5.Department of Mathematics and StatisticsUniversity of LeobenLeobenAustria

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