Integrality relations for polygonal dissections

TL;DR

This paper establishes integrality relations for areas in polygonal dissections, revealing invariance under deformation and providing new insights into classical dissection problems and Monsky's theorem.

Contribution

It introduces new integrality relations for areas in trapezoid and parallelogram dissections, connecting them to invariance properties and classical dissection theorems.

Findings

Area relations are invariant under dissection deformation

Square cannot be dissected into an odd number of equal-area triangles

Area polynomials for parallelograms have leading coefficient ±1

Abstract

Given a trapezoid dissected into triangles, the area of any triangle determined by either diagonal of the trapezoid is integral over the ring generated by the areas of the triangles in the dissection. Given a parallelogram dissected into triangles, the area of any one of the triangles of the dissection is integral over the ring generated by the areas of the other triangles. In both cases, the integrality relations are invariant under deformation of the dissection. The trapezoid theorem implies and provides a new context for Monsky's Equidissection Theorem that a square cannot be dissected into an odd number of triangles of equal area. A corollary of these results is that the area polynomials for parallelograms introduced in previous work have all leading coefficients equal to $\pm 1$.