American Mathematical Monthly最新文献

{"title":"Ten Points on a Cubic","authors":"Will Traves, David Wehlau","doi":"10.1080/00029890.2023.2274240","DOIUrl":"https://doi.org/10.1080/00029890.2023.2274240","url":null,"abstract":"AbstractThe 16-year old Blaise Pascal found an incidence relation that holds when six points lie on a conic. A century later, Braikenridge and Maclaurin extended Pascal’s result to a straightedge construction that characterizes when six points lie on a conic. Nearly 400 years later, we develop a straightedge construction to check whether ten points lie on a cubic curve.MSC: 14H5051A20 AcknowledgmentWe thank Bernd Sturmfels for suggesting the problem to us and Mike Roth for helpful discussions. We are grateful to J. Chris Fisher for suggesting an alternate approach to Construction 2 and for encouraging feedback. The computer algebra system MAGMA [Citation4] was extremely helpful and all figures in the paper were produced using GeoGebra [Citation16], which is a great resource for developing geometric intuition. We thank the members of the Editorial Board of the MONTHLY for their suggestions and advice. We also thank the anonymous referees for carefully reading the manuscript and providing many helpful suggestions.Notes1 Apparently, Hadamard was paraphrasing Paul Painlevé [Citation19], the French mathematician and statesman who served twice as Minister of War and twice as Prime Minister of France.2 There seems to be some controversy about the spelling of Steiner’s first name. The authoritative version of his collected works [Citation23] gives the author’s name spelled with a k and the subject’s name spelled with a c. Perhaps this confusion is common among people whose work is important enough to be translated into many languages.3 https://www.cut-the-knot.org/pythagoras/PPower.shtmlAdditional informationNotes on contributorsWill TravesWILL TRAVES(https://orcid.org/0000-0002-8115-1243) nearly failed freshman physics at Queen’s University but went on to become a professor and past chair of the Naval Academy mathematics department. His research interests include geometry, data science, and operations research.United States Naval Academy, Mail Stop 9E, Annapolis, MD 21402, USAtraves@usna.eduDavid WehlauDavid Wehlau(https://orcid.org/0000-0002-0272-8404) received his Ph.D. from Brandeis University and is a professor and past head of the Department of Mathematics and Computer Science at the Royal Military College of Canada. He is also a professor at Queen’s University and enjoys being able to work with mathematics students at both institutions.Royal Military College of Canada, PO Box 17000 Stn Forces, Kingston, ON, K7K 7B4, Canadawehlau@rmc.ca","PeriodicalId":7761,"journal":{"name":"American Mathematical Monthly","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-11-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"134957443","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"From Abel’s Binomial Theorem to Cayley’s Tree Formula","authors":"Marc Zucker","doi":"10.1080/00029890.2023.2276637","DOIUrl":"https://doi.org/10.1080/00029890.2023.2276637","url":null,"abstract":"AbstractWe derive Abel’s generalization of the binomial theorem and use it to present a short proof of Cayley’s theorem on the number of trees on n labeled vertices.MSC: 05C30 DISCLOSURE STATEMENTNo potential conflict of interest was reported by the author(s).","PeriodicalId":7761,"journal":{"name":"American Mathematical Monthly","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-11-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"134957591","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Abbott Dimension, Mathematics Inspired by <i>Flatland</i>","authors":"Jeremy Siegert","doi":"10.1080/00029890.2023.2275997","DOIUrl":"https://doi.org/10.1080/00029890.2023.2275997","url":null,"abstract":"AbstractWhat is the “right way” to define dimension? Mathematicians working in the early and middle 20th-century formalized three intuitive definitions of dimension that all turned out to be equivalent on separable metric spaces. But were these definitions the “right” ones? What would it mean to have the “right” definition of dimension? In this paper we attempt to inspire thought about these questions by introducing Abbott dimension, a geometrically intuitive definition of dimension based on Edwin Abbott’s 1884 novella Flatland. We show that while Abbott dimension has intuitive appeal, it does not always agree with the classical definitions of dimension on separable metric spaces.MSC: 54F45 AcknowledgmentThe author would like to thank Kate Franklin for her reading of, and input on, early drafts of this paper. The author would also like to thank the referees and editors for this paper whose input greatly improved it. This work was supported by the Israel Science Foundation grant No. 2196/20Notes1 Alternative formulations of what a separator could be do exist. Indeed, in what was perhaps the first precise formulation of dimension, Brouwer in [Citation5] defined his invariant, the dimensiongrad, of a space. This invariant is identical in definition to the large inductive dimension we have defined with the difference being that instead of using separators it uses the notion of a cut. A cut in a space X between disjoint closed subsets A and B is a third closed set C⊆X that is disjoint from A and B and is such that any continuum K⊆X that intersects A and B must also intersect C. This definition of dimension agrees with the classical definitions mentioned in the introduction on the class of compact metrizable spaces (see [Citation7]).2 Interestingly, for the classical definitions of dimension the result does not hold if the space is infinite dimensional. In [Citation8] a compact metric space of infinite dimension (with respect to any of the classical definitions) is constructed such that every subspace is either also infinite dimensional, or zero dimensional.3 In 1920 Knaster and Kuratowski asked in [Citation12] if a nondegenerate homogeneous continuum in the plane must be a closed curve. The next year Mazurkiewicz asked in [Citation13] if every continuum in the plane which is homeomorphic to all of its nondegenerate subcontinua must be an arc. In [Citation11] Knaster would give an example of a hereditarily indecomposable continuum in 1923. In [Citation2] Bing would answer Knaster and Kuratowski’s question negatively with his own construction in 1948, and Moise would answer Mazurkiewicz’s question negatively in [Citation14]. Moise dubbed his example the “pseudoarc” due to its similarity to an arc. Bing would go on in [Citation3] to show that his, Knaster’s, and Moise’s examples were all homeomorphic. This history and the results surrounding the pseudoarc can be found in [Citation15].4 One may denote the set of all continua in Rn by C(Rn) and endow it wi","PeriodicalId":7761,"journal":{"name":"American Mathematical Monthly","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-11-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136229853","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"The Factor Ring Structure of Quadratic Principal Ideal Domains","authors":"John Greene, Weizhi Jing","doi":"10.1080/00029890.2023.2261827","DOIUrl":"https://doi.org/10.1080/00029890.2023.2261827","url":null,"abstract":"AbstractPrevious authors have classified the possible factor rings of the Gaussian integers and the Eisenstein integers. Here, we extend this classification to the ring of integers of any quadratic number field, provided the ring has unique factorization. In the case of imaginary quadratic fields, the classification has exactly the same flavor as that of the Gaussian integers and Eisenstein integers. For real quadratic fields, the classification is only slightly more complicated.MSC: 11R1113F10 AcknowledgmentWe thank the anonymous referees for feedback which considerably improved the exposition of this article. We also thank the editors for their sharp eyes in proofing this article and for their help during the review process.Additional informationNotes on contributorsJohn GreeneJohn Greene received his Ph.D. from the University of Minnesota in 1983. He is a professor of mathematics at the University of Minnesota Duluth, where he has been for 30+ years. His research interests include special functions, combinatorics and (elementary) computational number theory.Department of Mathematics and Statistics, University of Minnesota Duluth, Duluth, MN 55812jgreene@d.umn.eduWeizhi JingWeizhi Jing received his B.S. in mathematics as part of a joint degree program between the University of Minnesota Duluth and Shanxi University of China in 2019 and received his Master’s degree from UMD in 2021. He is dedicated to mathematical education and hopes to pursue a Ph.D. in ring theory or algebraic number theory.Department of Mathematics and Statistics, University of Minnesota Duluth, Duluth, MN 55812jing0049@d.umn.edu","PeriodicalId":7761,"journal":{"name":"American Mathematical Monthly","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-11-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135136642","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"100 Years Ago This Month in <i>The American Mathematical Monthly</i>","authors":"Vadim Ponomarenko","doi":"10.1080/00029890.2023.2275985","DOIUrl":"https://doi.org/10.1080/00029890.2023.2275985","url":null,"abstract":"\"100 Years Ago This Month in The American Mathematical Monthly.\" The American Mathematical Monthly, ahead-of-print(ahead-of-print), p. 1","PeriodicalId":7761,"journal":{"name":"American Mathematical Monthly","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-11-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135590161","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Yet Another Integral Representation of Catalan’s Constant","authors":"Raymond Mortini","doi":"10.1080/00029890.2023.2274242","DOIUrl":"https://doi.org/10.1080/00029890.2023.2274242","url":null,"abstract":"","PeriodicalId":7761,"journal":{"name":"American Mathematical Monthly","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-11-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135679025","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"The Discrete Theorema Egregium","authors":"Thomas F. Banchoff, Felix Günther","doi":"10.1080/00029890.2023.2263299","DOIUrl":"https://doi.org/10.1080/00029890.2023.2263299","url":null,"abstract":"AbstractIn 1827, Gauss proved that Gaussian curvature is actually an intrinsic quantity, meaning that it can be calculated just from measurements within the surface. Before, curvature of surfaces could only be computed extrinsically, meaning that an ambient space is needed. Gauss named this remarkable finding Theorema Egregium. In this paper, we discuss a discrete version of this theorem for polyhedral surfaces. We give an elementary proof that the common extrinsic and intrinsic definitions of discrete Gaussian curvature are equivalent. AcknowledgmentThe authors thank the two anonymous reviewers and the editorial board for their valuable remarks. The research was initiated during the second author’s stay at the Max Planck Institute for Mathematics in Bonn. It was funded by the Deutsche Forschungsgemeinschaft DFG through the Collaborative Research Center TRR 109 “Discretization in Geometry and Dynamics” and under Germany’s Excellence Strategy – The Berlin Mathematics Research Center MATH+ (EXC-2046/1, project ID: 390685689).Additional informationNotes on contributorsThomas F. BanchoffTHOMAS BANCHOFF is professor emeritus of mathematics at Brown University, where he taught from 1967 to 2015. He received his Ph.D. from the University of California, Berkeley, in 1964 and served as president of the MAA from 1999 to 2000.Department of Mathematics, Brown University, Box 1917, 151 Thayer Street, Providence RI 02912Thomas_Banchoff@brown.eduFelix GüntherFELIX GÜNTHER received his Ph.D. in mathematics from Technische Universität Berlin in 2014. After holding postdoctoral positions at the Institut des Hautes Etudes Scientifiques in Bures-sur-Yvette, the Isaac Newton Institute for Mathematical Sciences in Cambridge, the Erwin Schrödinger International Institute for Mathematics and Physics in Vienna, the Max Planck Institute for Mathematics in Bonn, and the University of Geneva, he came back to Technische Universität Berlin in 2018. His research interests include discrete differential geometry and discrete complex analysis. He also has a passion for science communication.Technische Universität Berlin, Institut für Mathematik MA 8-3, Straße des 17. Juni 136, 10623 Berlin, Germanyfguenth@math.tu-berlin.de","PeriodicalId":7761,"journal":{"name":"American Mathematical Monthly","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-10-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135618674","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"The Hidden Twin of Morley’s Five Circles Theorem","authors":"Lorenz Halbeisen, Norbert Hungerbühler, Vanessa Loureiro","doi":"10.1080/00029890.2023.2254179","DOIUrl":"https://doi.org/10.1080/00029890.2023.2254179","url":null,"abstract":"We give an algebraic proof of a slightly extended version of Morley’s Five Circles Theorem. The theorem holds in all Miquelian Möbius planes obtained from a separable quadratic field extension, in particular in the classical real Möbius plane. Moreover, the calculations bring to light a hidden twin of the Five Circles Theorem that seems to have been overlooked until now.","PeriodicalId":7761,"journal":{"name":"American Mathematical Monthly","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-10-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135617924","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Commutators in the Rubik’s Cube Group","authors":"Timothy Sun","doi":"10.1080/00029890.2023.2263158","DOIUrl":"https://doi.org/10.1080/00029890.2023.2263158","url":null,"abstract":"Since the Rubik’s Cube was introduced in the 1970s, mathematicians and puzzle enthusiasts have studied the Rubik’s Cube group, i.e., the group of all ≈4.3×1019 solvable positions of the Rubik’s Cube. Group-theoretic ideas have found their way into practical methods for solving the Rubik’s Cube, and perhaps the most notable of these is the commutator. It is well-known that the commutator subgroup of the Rubik’s Cube group has index 2 and consists of the positions reachable by an even number of quarter turns. A longstanding open problem, first posed in 2004, asks whether every element of the commutator subgroup is itself a commutator. We answer this in the affirmative and sketch a generalization to the n×n×n Rubik’s Cube, for all n≥2.","PeriodicalId":7761,"journal":{"name":"American Mathematical Monthly","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-10-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135567955","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Reviews <i>Political Geometry: Rethinking Redistricting in the US with Math, Law, and Everything In Between</i> . Edited by Moon Duchin and Olivia Walch, Birkhäuser, 2022. 477 pp., ISBN 978-3319691602, $32.99.","authors":"Beth Malmskog","doi":"10.1080/00029890.2023.2266981","DOIUrl":"https://doi.org/10.1080/00029890.2023.2266981","url":null,"abstract":"Click to increase image sizeClick to decrease image size Notes1 If this real-life example doesn’t convince you, consider an imaginary state where 51% of the voters favor party A and 49% favor party B. If the voters of both parties are distributed uniformly across the state, every voting precinct will have 51% party A and 49% party B, and every possible district that can be drawn will have 51% party A and 49% party B. Thus party A will win ALL the districts, no matter how they are drawn.","PeriodicalId":7761,"journal":{"name":"American Mathematical Monthly","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-10-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135617566","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}