{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,7,30]],"date-time":"2025-07-30T16:43:47Z","timestamp":1753893827289,"version":"3.41.2"},"reference-count":0,"publisher":"The Electronic Journal of Combinatorics","issue":"1","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Electron. J. Combin."],"abstract":"<jats:p>In 1970 P. Monsky showed that a square cannot be triangulated into an odd number of triangles of equal areas; further, in 1990 E. A. Kasimatis and S. K. Stein proved that the trapezoid $T(\\alpha)$ whose vertices have the coordinates $(0,0)$, $(0,1)$, $(1,0)$, and $(\\alpha,1)$ cannot be triangulated into any number of triangles of equal areas if $\\alpha&gt;0$ is transcendental. In this paper we first establish a new asymptotic upper bound for the minimal difference between the smallest and the largest area in triangulations of a square into an odd number of triangles. More precisely, using some techniques from the theory of continued fractions, we construct a sequence of triangulations $T_{n_i}$ of the unit square into $n_i$ triangles, $n_i$ odd, so that the difference between the smallest and the largest area in $T_{n_i}$ is $O\\big(\\frac{1}{n_i^3}\\big)$. We then prove that for an arbitrarily fast-growing function $f:\\mathbb{N}\\to \\mathbb{N}$, there exists a transcendental number $\\alpha&gt;0$ and a sequence  of triangulations $T_{n_i}$ of the trapezoid $T(\\alpha)$ into $n_i$ triangles, so that the difference between the smallest and the largest area in $T_{n_i}$ is $O\\big(\\frac{1}{f(n_i)}\\big)$.<\/jats:p>","DOI":"10.37236\/624","type":"journal-article","created":{"date-parts":[[2020,1,11]],"date-time":"2020-01-11T03:44:29Z","timestamp":1578714269000},"source":"Crossref","is-referenced-by-count":0,"title":["On the Area Discrepancy of Triangulations of Squares and Trapezoids"],"prefix":"10.37236","volume":"18","author":[{"given":"Bernd","family":"Schulze","sequence":"first","affiliation":[]}],"member":"23455","published-online":{"date-parts":[[2011,7,1]]},"container-title":["The Electronic Journal of Combinatorics"],"original-title":[],"link":[{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v18i1p137\/pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v18i1p137\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2020,1,17]],"date-time":"2020-01-17T23:08:30Z","timestamp":1579302510000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/view\/v18i1p137"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2011,7,1]]},"references-count":0,"journal-issue":{"issue":"1","published-online":{"date-parts":[[2011,1,5]]}},"URL":"https:\/\/doi.org\/10.37236\/624","relation":{},"ISSN":["1077-8926"],"issn-type":[{"type":"electronic","value":"1077-8926"}],"subject":[],"published":{"date-parts":[[2011,7,1]]},"article-number":"P137"}}