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

TL;DR

This paper extends Monsky's theorem to provide lower bounds on area differences in square dissections into an odd number of triangles and introduces superpolynomial upper bounds using the Thue-Morse sequence.

Contribution

It generalizes Monsky's theorem to constrained maps and applies semi-algebraic geometry to establish exponential lower bounds, also providing the first superpolynomial upper bounds.

Findings

Lower bounds decrease doubly-exponentially with the number of triangles.

Explicit construction achieves superpolynomial upper bounds.

Application of semi-algebraic geometry to geometric dissection problems.

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.