Geodesic
On equidissection of balanced polygons
Zapiski Nauchnykh Seminarov POMI, Representation theory, dynamical systems, combinatorial methods. Part XXI, Tome 403 (2012), pp. 142-157 Cet article a éte moissonné depuis la source Math-Net.Ru

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In this paper we show that a lattice balanced polygon of odd area cannot be cut into an odd number of triangles of equal areas. The first result of this type was obtained by Paul Monsky in 1970. He proved that a square cannot be cut into an odd number of triangles of equal areas. In 2000, Sherman Stein conjectured that the same holds for any balanced polygon. We also show connections between the equidissection problem and tropical geometry. 
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D. Rudenko. On equidissection of balanced polygons. Zapiski Nauchnykh Seminarov POMI, Representation theory, dynamical systems, combinatorial methods. Part XXI, Tome 403 (2012), pp. 142-157. http://geodesic.mathdoc.fr/item/ZNSL_2012_403_a9/

[1] B. M. Bekker, N. Yu. Netsvetaev, “Generalized Sperner lemma and subdivisions into simplices of equal volume”, J. Math. Sci., 91:6 (1998), 3492–3498 | DOI | MR

[2] S. Lang, Algebra, Addison-Wesley, Reading, Massachusetts, 1965 | MR

[3] P. Monsky, “On dividing a square into triangles”, Amer. Math. Monthly, 77 (1970), 161–164 | DOI | MR | Zbl

[4] P. Monsky, “A conjecture of Stein on plane dissections”, Math. Z., 205 (1990), 583–592 | DOI | MR | Zbl

[5] I. Praton, “Cutting polyominos into equal-area triangles”, Amer. Math. Monthly, 109 (2002), 818–826 | DOI | MR | Zbl

[6] S. Stein, “A generalized conjecture about cutting a polygon into triangles of equal areas”, Discrete Comput. Geom., 24 (2000), 141–145 | DOI | MR | Zbl

[7] S. Stein, “Cutting a polygon into triangles of equal areas”, Math. Intelligencer, 26:1 (2004), 17–21 | DOI | MR | Zbl

[8] S. Stein, “Cutting a polyomino into triangles of equal areas”, Amer. Math. Monthly, 106 (1999), 255–257 | DOI | MR

[9] A. Hales, E. Straus, “Projective colorings”, Pacific J. Math., 99:1 (1982), 31–43 | DOI | MR | Zbl