Advertisement

Sequentially Divisible Dissections of Simple Polygons

  • Jin Akiyama
  • Toshinori Sakai
  • Jorge Urrutia
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 2098)

Abstract

A k-dissection D of a polygon P, is a partition of P into a set pf subpolygons {Q 1,...,Q m } with disjoint interiors such that these can be reassembled to form k polygons P 1,...,P k all similar to P. If for every j, 1 ≤ jk, the pieces of D can be assembled into j polygons, all similar to P, then D is called a sequentially k-divisible dissection. In this paper we show that any convex n-gon, n ≤ 5, has a sequentially k-divisible dissection with (k - 1)n+1 pieces. We give sequentially k-divisible dissections for some regular polygons with n ≥ 6 vertices. Furthermore, we show that any simple polygon P with n vertices has a (3k+4)-dissection with (2n - 2) +k(2n + ⌊ n/3 ⌋ - 4) pieces, k ≤ 0, that can be reassembled to form 4,7,..., or 3k + 4 polygons similar to P. We give similar results for star shaped polygons.

Keywords

Problem Complexity Computer Graphic Algorithm Analysis Discrete Mathematic Regular Polygon 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Akiyama, J., Nakamura, G.: An efficient dissection for a sequentially n-divisible square. Proc. of Discrete and Computational Geometry Workshop, Tokai University (1997), 80–89Google Scholar
  2. 2.
    Akiyama, J., Nakamura, G., Nozaki, A., Ozawa, K.: A note on the purely recursive dissection for a sequentially n-divisible square. in this proceedings.Google Scholar
  3. 3.
    Akiyama, J., Nakamura, G., Nozaki, A., Ozawa, K., Sakai, T.: The optimality of a certain purely recursive dissection for a sequentially n-divisible square. To appear in Computational Geometry; Theory and ApplicationsGoogle Scholar
  4. 4.
    Busschop, P.: Problèmes de géométrie. Nouvelle Correspondence Mathématique 2 (1876) 83–84Google Scholar
  5. 5.
    Collison, D. M.: Rational geometric dissections of convex polygons. Journal of Recreational Mathamatics 12(2) (1979-1980) 95–103.MathSciNetGoogle Scholar
  6. 6.
    Fourrey, E.: Curiosités Géométriques. Paris: Vuibert et Nony (1907)Google Scholar
  7. 7.
    Frederickson, G. N.: Dissections: Plane & Fancy. Cambridge University Press (1997)Google Scholar
  8. 8.
    Gardener, M.: The 2nd. Scientific American Book of Mathematical Puzzles and Diversions. New York: Simon and Schuster (1961)Google Scholar
  9. 9.
    Kraitchik, M.: Mathematical Recreations. New York: Northon (1942)Google Scholar
  10. 10.
    Lindgren, H.: Geometric Dissections. Princeton, N.J.: D. Van Nostrand Company (1964)Google Scholar
  11. 11.
    Nozaki, A.: On the dissection of a square into squares (in Japanese). Suugaku-Semina No.12 (1999) 52–56Google Scholar
  12. 12.
    Ozawa, K.: Entertainer in a classroom (in Japanese). Suugaku-Seminar No.10 (1988) cover pageGoogle Scholar
  13. 13.
    Urrutia, J.: Art Gallery and Illumination Problems in Handbook on Computational Geometry. Sack, J.R. and Urrutia, J. eds. Elsevier Science Publishers (2000)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2001

Authors and Affiliations

  • Jin Akiyama
    • 1
  • Toshinori Sakai
    • 2
  • Jorge Urrutia
    • 3
  1. 1.Research Institute of Educational DevelopmentTokai UniversityTokyoJapan
  2. 2.Research Institute of EducationTokai UniversityTokyoJapan
  3. 3.Instituto de MatemáticasCiudad Universitaria Universidad Nacional Autónoma de MéxicoMéxico D.F.México

Personalised recommendations