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arXiv · 1708.02891

Area difference bounds for dissections of a square into an odd number of triangles

Jean-Philippe Labbé·Günter Rote·Günter M. Ziegler
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Abstract

Monsky's theorem from 1970 states that a square cannot be dissected into an odd number of triangles of the same area, but it does not give a lower bound for the area differences that must occur. We extend Monsky's theorem to "constrained framed maps"; based on this we can apply a gap theorem from semi-algebraic geometry to a polynomial area difference measure and thus get a lower bound for the area differences that decreases doubly-exponentially with the number of triangles. On the other hand, we obtain the first superpolynomial upper bounds for this problem, derived from an explicit construction that uses the Thue-Morse sequence.

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Jean-Philippe Labbé, Günter Rote, Günter M. Ziegler. 2017-08-09. Area difference bounds for dissections of a square into an odd number of triangles. https://doi.org/10.1080/10586458.2018.1459961

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