{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,7,30]],"date-time":"2025-07-30T16:43:44Z","timestamp":1753893824229,"version":"3.41.2"},"reference-count":0,"publisher":"The Electronic Journal of Combinatorics","issue":"3","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Electron. J. Combin."],"abstract":"<jats:p>A $d$-dimensional simplex $S$ is called a $k$-reptile (or a $k$-reptile simplex) if it can be tiled by $k$ simplices with disjoint interiors that are all mutually congruent and similar to $S$. For $d=2$, triangular $k$-reptiles exist for all $k$ of the form $a^2, 3a^2$ or $a^2 + b^2$ and they have been completely characterized by Snover, Waiveris, and Williams. On the other hand, the only $k$-reptile simplices that are known for $d \\ge 3$, have $k = m^d$, where $m$ is a positive integer. We substantially simplify the proof by Matou\u0161ek and the second author that for $d=3$, $k$-reptile tetrahedra can exist only for $k=m^3$. We then prove a weaker analogue of this result for $d=4$ by showing that four-dimensional $k$-reptile simplices can exist only for $k=m^2$.<\/jats:p>","DOI":"10.37236\/6113","type":"journal-article","created":{"date-parts":[[2020,1,10]],"date-time":"2020-01-10T15:51:42Z","timestamp":1578671502000},"source":"Crossref","is-referenced-by-count":0,"title":["On the Nonexistence of $k$-Reptile Simplices in $\\mathbb R^3$ and $\\mathbb R^4$"],"prefix":"10.37236","volume":"24","author":[{"given":"Jan","family":"Kyn\u010dl","sequence":"first","affiliation":[]},{"given":"Zuzana","family":"Pat\u00e1kov\u00e1","sequence":"additional","affiliation":[]}],"member":"23455","published-online":{"date-parts":[[2017,7,14]]},"container-title":["The Electronic Journal of Combinatorics"],"original-title":[],"link":[{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v24i3p1\/pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v24i3p1\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2020,1,17]],"date-time":"2020-01-17T04:55:40Z","timestamp":1579236940000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/view\/v24i3p1"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2017,7,14]]},"references-count":0,"journal-issue":{"issue":"3","published-online":{"date-parts":[[2017,7,14]]}},"URL":"https:\/\/doi.org\/10.37236\/6113","relation":{},"ISSN":["1077-8926"],"issn-type":[{"type":"electronic","value":"1077-8926"}],"subject":[],"published":{"date-parts":[[2017,7,14]]},"article-number":"P3.1"}}