{"title":"用边长为 $$n^{-1/2-\\epsilon }$ 的等边三角形完美包装等边三角形","authors":"Janusz Januszewski, Łukasz Zielonka","doi":"10.1007/s00454-024-00654-w","DOIUrl":null,"url":null,"abstract":"<p>Equilateral triangles of sidelengths 1, <span>\\(2^{-t}\\)</span>, <span>\\(3^{-t}\\)</span>, <span>\\(4^{-t},\\ldots \\ \\)</span> can be packed perfectly into an equilateral triangle, provided that <span>\\(\\ 1/2<t \\le 37/72\\)</span>. Moreover, for <i>t</i> slightly greater than 1/2, squares of sidelengths 1, <span>\\(2^{-t}\\)</span>, <span>\\(3^{-t}\\)</span>, <span>\\(4^{-t},\\ldots \\ \\)</span> can be packed perfectly into a square <span>\\(S_t\\)</span> in such a way that some squares have a side parallel to a diagonal of <span>\\(S_t\\)</span> and the remaining squares have a side parallel to a side of <span>\\(S_t\\)</span>.</p>","PeriodicalId":0,"journal":{"name":"","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-05-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Perfectly Packing an Equilateral Triangle by Equilateral Triangles of Sidelengths $$n^{-1/2-\\\\epsilon }$$\",\"authors\":\"Janusz Januszewski, Łukasz Zielonka\",\"doi\":\"10.1007/s00454-024-00654-w\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<p>Equilateral triangles of sidelengths 1, <span>\\\\(2^{-t}\\\\)</span>, <span>\\\\(3^{-t}\\\\)</span>, <span>\\\\(4^{-t},\\\\ldots \\\\ \\\\)</span> can be packed perfectly into an equilateral triangle, provided that <span>\\\\(\\\\ 1/2<t \\\\le 37/72\\\\)</span>. Moreover, for <i>t</i> slightly greater than 1/2, squares of sidelengths 1, <span>\\\\(2^{-t}\\\\)</span>, <span>\\\\(3^{-t}\\\\)</span>, <span>\\\\(4^{-t},\\\\ldots \\\\ \\\\)</span> can be packed perfectly into a square <span>\\\\(S_t\\\\)</span> in such a way that some squares have a side parallel to a diagonal of <span>\\\\(S_t\\\\)</span> and the remaining squares have a side parallel to a side of <span>\\\\(S_t\\\\)</span>.</p>\",\"PeriodicalId\":0,\"journal\":{\"name\":\"\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":0.0,\"publicationDate\":\"2024-05-11\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://doi.org/10.1007/s00454-024-00654-w\",\"RegionNum\":0,\"RegionCategory\":null,\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"\",\"JCRName\":\"\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1007/s00454-024-00654-w","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
Perfectly Packing an Equilateral Triangle by Equilateral Triangles of Sidelengths $$n^{-1/2-\epsilon }$$
Equilateral triangles of sidelengths 1, \(2^{-t}\), \(3^{-t}\), \(4^{-t},\ldots \ \) can be packed perfectly into an equilateral triangle, provided that \(\ 1/2<t \le 37/72\). Moreover, for t slightly greater than 1/2, squares of sidelengths 1, \(2^{-t}\), \(3^{-t}\), \(4^{-t},\ldots \ \) can be packed perfectly into a square \(S_t\) in such a way that some squares have a side parallel to a diagonal of \(S_t\) and the remaining squares have a side parallel to a side of \(S_t\).