{"title":"欧拉和福斯定理的高维版本","authors":"Peter Gibson, Nicolau Saldanha, Carlos Tomei","doi":"10.1007/s40598-023-00243-4","DOIUrl":null,"url":null,"abstract":"<div><p>We present higher dimensional versions of the classical results of Euler and Fuss, both of which are special cases of the celebrated Poncelet porism. Our results concern polytopes, specifically simplices, parallelotopes and cross polytopes, inscribed in a given ellipsoid and circumscribed to another. The statements and proofs use the language of linear algebra. Without loss, one of the ellipsoids is the unit sphere and the other one is also centered at the origin. Let <i>A</i> be the positive symmetric matrix taking the outer ellipsoid to the inner one. If <span>\\({\\text {tr}}\\, A = 1\\)</span>, there exists a bijection between the orthogonal group <i>O</i>(<i>n</i>) and the set of such labeled simplices. Similarly, if <span>\\({\\text {tr}}\\, A^2 = 1\\)</span>, there are families of parallelotopes and of cross polytopes, also indexed by <i>O</i>(<i>n</i>).</p></div>","PeriodicalId":37546,"journal":{"name":"Arnold Mathematical Journal","volume":null,"pages":null},"PeriodicalIF":0.0000,"publicationDate":"2024-01-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Higher Dimensional Versions of Theorems of Euler and Fuss\",\"authors\":\"Peter Gibson, Nicolau Saldanha, Carlos Tomei\",\"doi\":\"10.1007/s40598-023-00243-4\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<div><p>We present higher dimensional versions of the classical results of Euler and Fuss, both of which are special cases of the celebrated Poncelet porism. Our results concern polytopes, specifically simplices, parallelotopes and cross polytopes, inscribed in a given ellipsoid and circumscribed to another. The statements and proofs use the language of linear algebra. Without loss, one of the ellipsoids is the unit sphere and the other one is also centered at the origin. Let <i>A</i> be the positive symmetric matrix taking the outer ellipsoid to the inner one. If <span>\\\\({\\\\text {tr}}\\\\, A = 1\\\\)</span>, there exists a bijection between the orthogonal group <i>O</i>(<i>n</i>) and the set of such labeled simplices. Similarly, if <span>\\\\({\\\\text {tr}}\\\\, A^2 = 1\\\\)</span>, there are families of parallelotopes and of cross polytopes, also indexed by <i>O</i>(<i>n</i>).</p></div>\",\"PeriodicalId\":37546,\"journal\":{\"name\":\"Arnold Mathematical Journal\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":0.0000,\"publicationDate\":\"2024-01-22\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Arnold Mathematical Journal\",\"FirstCategoryId\":\"1085\",\"ListUrlMain\":\"https://link.springer.com/article/10.1007/s40598-023-00243-4\",\"RegionNum\":0,\"RegionCategory\":null,\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q3\",\"JCRName\":\"Mathematics\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Arnold Mathematical Journal","FirstCategoryId":"1085","ListUrlMain":"https://link.springer.com/article/10.1007/s40598-023-00243-4","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q3","JCRName":"Mathematics","Score":null,"Total":0}
引用次数: 0
摘要
我们提出了欧拉和福斯经典结果的高维版本，这两个结果都是著名的庞斯莱孔主义的特例。我们的结果涉及多面体，特别是简面、平行多面体和交叉多面体，它们都刻在给定的椭球体上，并以另一个椭球体为圆心。陈述和证明使用线性代数语言。在不损失任何信息的情况下，其中一个椭圆体是单位球面，另一个椭圆体也以原点为中心。设 A 是将外椭球面取为内椭球面的正对称矩阵。如果 \({\text {tr}}\, A = 1\), 那么在正交群 O(n) 和这样的标注简约集之间存在一个双射。类似地，如果 \({\text {tr}}\, A^2 = 1\), 则存在平行多面体族和交叉多面体族，同样以 O(n) 为索引。
Higher Dimensional Versions of Theorems of Euler and Fuss
We present higher dimensional versions of the classical results of Euler and Fuss, both of which are special cases of the celebrated Poncelet porism. Our results concern polytopes, specifically simplices, parallelotopes and cross polytopes, inscribed in a given ellipsoid and circumscribed to another. The statements and proofs use the language of linear algebra. Without loss, one of the ellipsoids is the unit sphere and the other one is also centered at the origin. Let A be the positive symmetric matrix taking the outer ellipsoid to the inner one. If \({\text {tr}}\, A = 1\), there exists a bijection between the orthogonal group O(n) and the set of such labeled simplices. Similarly, if \({\text {tr}}\, A^2 = 1\), there are families of parallelotopes and of cross polytopes, also indexed by O(n).
期刊介绍：
The Arnold Mathematical Journal publishes interesting and understandable results in all areas of mathematics. The name of the journal is not only a dedication to the memory of Vladimir Arnold (1937 – 2010), one of the most influential mathematicians of the 20th century, but also a declaration that the journal should serve to maintain and promote the scientific style characteristic for Arnold''s best mathematical works. Features of AMJ publications include: Popularity. The journal articles should be accessible to a very wide community of mathematicians. Not only formal definitions necessary for the understanding must be provided but also informal motivations even if the latter are well-known to the experts in the field. Interdisciplinary and multidisciplinary mathematics. AMJ publishes research expositions that connect different mathematical subjects. Connections that are useful in both ways are of particular importance. Multidisciplinary research (even if the disciplines all belong to pure mathematics) is generally hard to evaluate, for this reason, this kind of research is often under-represented in specialized mathematical journals. AMJ will try to compensate for this.Problems, objectives, work in progress. Most scholarly publications present results of a research project in their “final'' form, in which all posed questions are answered. Some open questions and conjectures may be even mentioned, but the very process of mathematical discovery remains hidden. Following Arnold, publications in AMJ will try to unhide this process and made it public by encouraging the authors to include informal discussion of their motivation, possibly unsuccessful lines of attack, experimental data and close by research directions. AMJ publishes well-motivated research problems on a regular basis. Problems do not need to be original; an old problem with a new and exciting motivation is worth re-stating. Following Arnold''s principle, a general formulation is less desirable than the simplest partial case that is still unknown.Being interesting. The most important requirement is that the article be interesting. It does not have to be limited by original research contributions of the author; however, the author''s responsibility is to carefully acknowledge the authorship of all results. Neither does the article need to consist entirely of formal and rigorous arguments. It can contain parts, in which an informal author''s understanding of the overall picture is presented; however, these parts must be clearly indicated.
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