Monsky’s Theorem / O Teorema de Monsky

Authors

  • Guilherme Israel Vedana Brazilian Journals Publicações de Periódicos, São José dos Pinhais, Paraná
  • João Biesdorf
  • Ivan Italo Gonzales Gargate

DOI:

https://doi.org/10.34117/bjdv7n8-523

Keywords:

Dissection of a square into triangles of equal area, 2-adic valuation.

Abstract

The main objective of this paper is to prove Monsky’s Theorem, that provides a beautiful application of the 2-adic valuation in order to solve a plane geometry problem. This theorem states that given any dissection of a square into finitely many nonoverlapping triangles of equal area the number of triangles must be even. In order to prove this statement, we will need some previous results from Combinatorial Topology and Algebra.

 

References

[AZ14] M. AIGNER & G. M. ZIEGLER. Proofs from THE BOOK. Springer-Verlag, Berlin Heidelberg, 5th ed. 2014.

[CEPID15] CEPID CeMEAI. Íntegra: Seminário de Coisas Legais – números p-ádicos e Teorema de Monsky. 2015. Available in: https://www.youtube.com/watch?v=TO-SMI9tccU

[Co18] O. F. CONNOLLY. One Square and an Odd Number of Triangles. 2018. Available in: https://www.maths.tcd.ie/~vdots/teaching/files/MA341C-1819/341CR6-2.pdf

[EP05] A. J. ENGLER & A. PRESTEL. Valued Fields. Springer Monographs in Mathematics. Springer-Verlag Berlin Heidelberg. 2005.

[Ja75] N. JACOBSON: Lectures in Abstract Algebra, Part III: Theory of Fields and Galois Theory. Graduate Texts in Mathematics 32, Springer, New York 1975.

[Kas89] E. A. KASIMATIS. Dissections of Regular Polygons into Triangles of Equal Areas. Discrete and Comp. Geometry 4 (1989), 375-381.

[Kat00] S. KATOK. p-adic Analysis Compared with Real. 1st ed. American Mathematical Society – Student Mathematical Library, vol. 27, 2000.

[Me79] D. G. MEAD. Dissection of the Hypercube into Simplexes. Proceedings of the Amer. Math. Society. Vol. 76, Nº2, Sep. 1979.

[Mon70] P. MONSKY: On dividing a square into triangles. Amer. Math. Monthly 77 (1970), 161-164.

[Mon90] P. MONSKY. A conjecture of Stein on plane dissections. Math Z 205, 583–592 (1990).

[Mor16] A. F. MORAGUES. What is... Monsky’s Theorem? Available in: https://math.osu.edu/sites/math.osu.edu/files/ferre_MonskyThm_2016.pdf

[RT67] F. RICHMAN & J. THOMAS. Problem 5471, Amer. Math. Monthly 74 (1967), 329.

[St04] S. STEIN. Cutting a Polygon into Triangles of Equal Areas. The Mathematical Intelligencer 26, 17–21 (2004).

[Th68] J. THOMAS. A Dissection Problem. Mathematics Magazine, 41:4,

-190, 1968.

[Xu12] M. XU. Sperner’s Lemma. Available in: https://math.berkeley.edu/~moorxu/misc/equiareal.pdf

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Published

2021-08-23

How to Cite

Vedana, G. I., Biesdorf, J., & Gargate, I. I. G. (2021). Monsky’s Theorem / O Teorema de Monsky. Brazilian Journal of Development, 7(8), 83547–83557. https://doi.org/10.34117/bjdv7n8-523

Issue

Section

Original Papers