Area Difference Bounds for Dissections of a Square into an Odd Number of Triangles

ABSTRACT

Monsky’s theorem from 1970 states that a square cannot be dissected into an odd number n 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 n. 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.

KEYWORDS:

• Monsky’s theorem
• dissection
• algebraic gap theorems
• Thue-Morse sequence

2010 AMS Subject Classifcation:

• 52B45

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Correction

Acknowledgments

We are grateful to Moritz Firsching, Arnau Padrol, Francisco Santos, Raman Sanyal, and Louis Theran for their input, advice, and many valuable discussions.

Additional information

Funding

This research was supported by the DFG Collaborative Research Center TRR 109 “Discretization in Geometry and Dynamics.”