  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.
