Discrete & Computational Geometry

, Volume 4, Issue 4, pp 375–381

# Dissections of regular polygons into triangles of equal areas

• E. A. Kasimatis
Article

## Abstract

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.

## Keywords

Discrete Comput Geom Prime Divisor Affine Transformation Equal Area 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.

## References

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.
3. 3.
K. Ireland and M. Rosen,A Classical Introduction to Modern Number Theory, Springer-Verlag, New York, 1982.
4. 4.
D. G. Mead, Dissection of the hypercube into simplexes,Proc. Amer. Math. Soc.,76 (1979), 302–304.
5. 5.
P. Monsky, On dividing a square into triangles,Amer. Math. Monthly,77 (1970), 161–164.
6. 6.
F. Richman and J. Thomas, Problem 5479,Amer. Math. Monthly,74 (1967), 329.
7. 7.
J. Thomas, A dissection problem,Math. Mag.,41 (1968), 187–190.