{"title":"Rupert property of some particular n-simplices and n-octahedrons","authors":"Pongbunthit Tonpho, Wacharin Wichiramala","doi":"10.1007/s10711-024-00894-3","DOIUrl":null,"url":null,"abstract":"<p>Three hundred years ago, Prince Rupert of Rhine showed that a unit cube has the property that one copy of it can be passed through a suitable hole in another copy. Under this situation, we say that a unit cube has the Rupert property. In the past years, there are many research studying about the Rupert property of many solids in <span>\\(\\mathbb {R}^3\\)</span>. For higher dimensions, the <i>n</i>-dimensional cube and the regular <i>n</i>-simplex were studied to have the Rupert property. In this work, we focus on the Rupert property of some polyhedrons in <i>n</i> dimensions. In particular, we show that some particular <i>n</i>-dimensional simplices, generalized <i>n</i>-dimensional octahedrons and some related solids in <span>\\(\\mathbb {R}^n\\)</span> have the Rupert property using arbitrarily small rotations and translations.</p>","PeriodicalId":55103,"journal":{"name":"Geometriae Dedicata","volume":"33 1","pages":""},"PeriodicalIF":0.5000,"publicationDate":"2024-03-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Geometriae Dedicata","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1007/s10711-024-00894-3","RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q3","JCRName":"MATHEMATICS","Score":null,"Total":0}
引用次数: 0
Abstract
Three hundred years ago, Prince Rupert of Rhine showed that a unit cube has the property that one copy of it can be passed through a suitable hole in another copy. Under this situation, we say that a unit cube has the Rupert property. In the past years, there are many research studying about the Rupert property of many solids in \(\mathbb {R}^3\). For higher dimensions, the n-dimensional cube and the regular n-simplex were studied to have the Rupert property. In this work, we focus on the Rupert property of some polyhedrons in n dimensions. In particular, we show that some particular n-dimensional simplices, generalized n-dimensional octahedrons and some related solids in \(\mathbb {R}^n\) have the Rupert property using arbitrarily small rotations and translations.
三百年前，莱茵王子鲁珀特证明，一个单位立方体具有这样的性质：它的一个副本可以穿过另一个副本上的一个合适的孔。在这种情况下，我们说单位立方体具有鲁珀特性质。在过去的几年里，有许多关于 \(\mathbb {R}^3\) 中许多固体的鲁珀特性质的研究。对于更高的维度，人们研究了 n 维立方体和正 n 次方体具有鲁珀特性质。在这项工作中，我们将重点研究 n 维多面体的鲁珀特性质。特别是，我们证明了在\(\mathbb {R}^n\)中的一些特定的n维简面、广义n维八面体和一些相关的实体在任意小的旋转和平移下具有鲁珀特性质。
期刊介绍：
Geometriae Dedicata concentrates on geometry and its relationship to topology, group theory and the theory of dynamical systems.
Geometriae Dedicata aims to be a vehicle for excellent publications in geometry and related areas. Features of the journal will include:
A fast turn-around time for articles.
Special issues centered on specific topics.
All submitted papers should include some explanation of the context of the main results.
Geometriae Dedicata was founded in 1972 on the initiative of Hans Freudenthal in Utrecht, the Netherlands, who viewed geometry as a method rather than as a field. The present Board of Editors tries to continue in this spirit. The steady growth of the journal since its foundation is witness to the validity of the founder''s vision and to the success of the Editors'' mission.
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