Advertisement

The Dissection of Rectangles Into Squares

  • R. L. Brooks
  • C. A. B. Smith
  • A. H. Stone
  • W. T. Tutte
Part of the Modern Birkhäuser Classics book series (MBC)

Abstract

We consider the problem of dividing a rectangle into a finite number of non-overlapping squares, no two of which are equal. A dissection of a rectangle R into a finite number n of non-overlapping squares is called a squaring of R of order n; and the n squares are the elements of the dissection. The term “elements” is also used for the lengths of the sides of the elements. If there is more than one element and the elements are all unequal, the squaring is called perfect, and R is a perfect rectangle.

Keywords

Normal Form Equilateral Triangle Open Plane Vertical Side Horizontal Side 
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.

Bibliography

  1. 1.
    M. Abe, On the problem to cover simply and without gap the inside of a square with a finite number of squares which are all different from one another, Proceedings of the Physico-Mathematical Society of Japan, (3), vol. 14 (1932), pp. 385–387.Google Scholar
  2. 2.
    W. W. Rouse Ball, Mathematical Recreations, 11th ed., New York, 1939.Google Scholar
  3. 3.
    C. W. Borchardt, Ueber eine der Interpolation entsprechende Darstellung der Elimina-tions-Resultante, Journal fiir Mathematik, vol. 57 (1860), pp. 111–121.MATHGoogle Scholar
  4. 4.
    S. Chowla, Division of a rectangle into unequal squares, Mathematics Student, vol. 7 (1939), p. 69.Google Scholar
  5. 5.
    S. Chowla, ibid., Question 1779.Google Scholar
  6. 6.
    M. Dehn, Zerlegung von Rechtecke in Rechtecken, Mathematische Annalen, vol. 57 (1903), pp. 314-332.MATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    Jarenkiewycz, Zeitschrift für die mathematische und naturwissenschaftliche Unter-richt, vol. 66(1935), p. 251, Aufgabe 1242; and solution, op. cit., vol. 68(1937), p. 43. (Also solutions by Mahrenholz and Sprague.)Google Scholar
  8. 8.
    J. H. Jeans, The Mathematical Theory of Electricity and Magnetism, Cambridge, 1908.Google Scholar
  9. 9.
    G. Kirchhoff, Ueber die Auflb’sung der Gleichungen, auf welche man bei der Unter-suchung der linearen Vertheilung galvanischer Strome gefiihrt wird, Annalen d. Physik und Chemie, vol. 72 (1847), p. 497.CrossRefGoogle Scholar
  10. 10.
    M. Kraitchik, La Mathematique des Jeux, Brussels, 1930.Google Scholar
  11. 11.
    Z. MoroŃ, Przegląd Mat. Fiz., vol. 3 (1925), pp. 152, 153.Google Scholar
  12. 12.
    A. Schoenflies-M. Dehn, Einfuehrung in die analytische Geometrie der Ebene und des Raumes, 2d ed., Berlin, 1931.Google Scholar
  13. 13.
    R. Sprague, Mathematische Zeitschrift, vol. 45 (1939), p. 607.CrossRefMathSciNetGoogle Scholar
  14. 14.
    H. Steinhaus, Mathematical Snapshots,New York, 1938.Google Scholar
  15. 15.
    A. Stöhr, Zerlegung von Rechtecken in inkongruente Quadrate, Thesis, Berlin.Google Scholar
  16. 16.
    A. H. Stone, Question E. 401 and solution, American Mathematical Monthly, vol. 47 (1940).Google Scholar
  17. 17.
    H. Toepken, Aufgabe 242, Jahresberichte der deutschen Mathematiker-Vereinigung, vol. 47 (1937), p. 2. Google Scholar
  18. 18.
    H. Toepken, Aufgabe 271, op. cit., vol. 48 (1938), p. 73. Google Scholar
  19. 19.
    H. W. Turnbull, Theory of Matrices, Determinants, and Invariants,London, 1929.Google Scholar

Copyright information

© Birkhäuser Boston, a part of Springer Science+Business Media, LLC 2009

Authors and Affiliations

  • R. L. Brooks
    • 1
  • C. A. B. Smith
    • 1
  • A. H. Stone
    • 1
  • W. T. Tutte
    • 1
  1. 1.Trinity College, Princeton UniversityCambridgeEngland

Personalised recommendations