## Jean-Philippe Labbé, Günter Rote, and Günter M. Ziegler:

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

Preprint, August 2017, 29 pages, arXiv:1708.02891 [math.MG].
### 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.

Last update: August 10, 2017.