Discrete & Computational Geometry

, Volume 4, Issue 4, pp 375–381 | Cite as

Dissections of regular polygons into triangles of equal areas

  • E. A. Kasimatis


This paper answers the question, “If a regular polygon withn sides is dissected intom triangles of equal areas, mustm be a multiple ofn?” Forn=3 the answer is “no,” since a triangle can be cut into any positive integral number of triangles of equal areas. Forn=4 the answer is again “no,” since a square can be cut into two triangles of equal areas. However, Monsky showed that a square cannot be dissected into an odd number of triangles of equal areas.

We show that ifn is at least 5, then the answer is “yes.” Our approach incorporates the techniques of Thomas, Monsky, and Mead, in particular, the use of Sperner's lemma and non-Archimedean valuations, but also makes use of affine transformations to distort a given regular polygon into one to which those techniques apply.


Discrete Comput Geom Prime Divisor Affine Transformation Equal Area Regular Polygon 
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  1. 1.
    G. Bachman,Introduction to p-adic Numbers and Valuation Theory, Academic Press, New York, 1964.Google Scholar
  2. 2.
    H. Hasse,Number Theory, Springer-Verlag, Berlin, 1980.CrossRefzbMATHGoogle Scholar
  3. 3.
    K. Ireland and M. Rosen,A Classical Introduction to Modern Number Theory, Springer-Verlag, New York, 1982.CrossRefzbMATHGoogle Scholar
  4. 4.
    D. G. Mead, Dissection of the hypercube into simplexes,Proc. Amer. Math. Soc.,76 (1979), 302–304.MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    P. Monsky, On dividing a square into triangles,Amer. Math. Monthly,77 (1970), 161–164.MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    F. Richman and J. Thomas, Problem 5479,Amer. Math. Monthly,74 (1967), 329.MathSciNetCrossRefGoogle Scholar
  7. 7.
    J. Thomas, A dissection problem,Math. Mag.,41 (1968), 187–190.MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer-Verlag New York Inc. 1989

Authors and Affiliations

  • E. A. Kasimatis
    • 1
  1. 1.Department of MathematicsCalifornia State UniversitySacramentoUSA

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