An illustrated encyclopedia of area relations

TL;DR

This paper investigates polynomial relations among triangle areas in triangulations of squares, showing finiteness of certain polynomial sets and providing an explicit area encyclopedia for up to four triangles.

Contribution

It introduces a finite set of irreducible polynomials related to triangle areas and computes this set for up to four triangles, advancing understanding of area relations in triangulations.

Findings

Finite set of irreducible polynomials for area relations

Explicit computation of area encyclopedia for l≤4

Polynomial constraints on triangle areas in square dissections

Abstract

To any combinatorial triangulation $T$ of a square, there is an associated polynomial relation $p_T$ among the areas of the triangles of $T$. With the goal of understanding this polynomial, we consider polynomials obtained from $p_T$ by choosing $l$ of its variables and specializing $p_T$ to these variables by zeroing out the remaining variables. We show that for fixed $l$, the set ${\mathcal E}_l$ of integer polynomials that appear as irreducible factors of such specializations is finite. We compute this area encyclopedia ${\mathcal E}_l$ for $l\leq 4$. We also show that in any dissection of a square into $l$ triangles, the areas of the triangles must satisfy a polynomial in ${\mathcal E}_l$. Our results are obtained by studying the rational map that associates to each drawing of $T$ the tuple of areas of the triangles in that drawing. By analyzing the ways of approaching the base locus, we derive restrictions on points of the closure of the image of this map.