Pub Date : 2023-11-14DOI: 10.1080/00029890.2023.2274240
Will Traves, David Wehlau
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
{"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}
Pub Date : 2023-11-14DOI: 10.1080/00029890.2023.2276637
Marc Zucker
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).
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Pub Date : 2023-11-14DOI: 10.1080/00029890.2023.2275997
Jeremy Siegert
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
定义维度的“正确方法”是什么?20世纪早期和中期的数学家们形式化了三种直观的维度定义，结果证明它们在可分离度量空间上是等价的。但这些定义是“正确的”吗?拥有“正确”的维度定义意味着什么?在本文中，我们试图通过引入艾伯特维度来启发人们对这些问题的思考。艾伯特维度是基于埃德温·艾伯特1884年的中篇小说《平地》而提出的一个几何上直观的维度定义。我们表明，虽然雅培维数具有直观的吸引力，但它并不总是符合可分离度量空间上维数的经典定义。作者要感谢Kate Franklin对本文早期草稿的阅读和投入。作者也要感谢审稿人和编辑，他们的投入大大改进了本文。这项工作得到了以色列科学基金会第2196/20号拨款的支持。实际上，在可能是第一个精确的维数表述中，browwer在[引文5]中定义了他的不变量，空间的维数梯度。这个不变量在定义上和我们定义的大归纳维是一样的不同之处在于它没有使用分隔符而是使用了切的概念。在不相交的闭子集A和B之间的空间X中的切是与A和B不相交的第三个闭集C和X，并且使得与A和B相交的任何连续体K和X也必须与C相交。这个维数的定义与紧度空间类的介绍中提到的经典定义一致(参见[引文7])有趣的是，对于维数的经典定义，如果空间是无限维，结果就不成立了。在[Citation8]中，构造了一个无限维的紧致度量空间(相对于任何经典定义)，使得每个子空间要么也是无限维的，要么是零维的1920年，Knaster和Kuratowski在[Citation12]中提出，平面上的非退化齐次连续体是否一定是闭合曲线。第二年，Mazurkiewicz在[Citation13]中问道，平面上与所有非简并次连续体同胚的连续体是否一定是弧。在1923年，Knaster给出了一个遗传上不可分解连续体的例子。在[Citation2]中，Bing在1948年用自己的结构否定地回答了Knaster和Kuratowski的问题，而Moise在[Citation14]中否定地回答了Mazurkiewicz的问题。Moise将他的例子称为“伪弧”，因为它与弧相似。Bing在[引文3]中继续证明了他、Knaster和Moise的例子都是同胚的。这段历史和围绕伪弧的结果可以在[Citation15]中找到我们可以用C(Rn)表示Rn中所有连续体的集合，并赋予它豪斯多夫度规dh。给定两个连续体X,Y∈C(Rn)，dh(X,Y)是所有≥0的最小值，使得X⊥B(Y， λ)和Y⊥B(X， λ)。这里B(X， λ)是半径为λ X的开度规球。我们引用的结果表明，遗传不可分解连续体是这个空间的密集子集。作者简介:jeremy Siegert jeremy Siegert在田纳西大学诺克斯维尔分校获得数学博士学位。他目前是以色列贝尔舍瓦内盖夫本古里安大学的博士后研究员。
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Pub Date : 2023-11-10DOI: 10.1080/00029890.2023.2261827
John Greene, Weizhi Jing
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
{"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}
Pub Date : 2023-11-06DOI: 10.1080/00029890.2023.2275985
Vadim Ponomarenko
"100 Years Ago This Month in The American Mathematical Monthly." The American Mathematical Monthly, ahead-of-print(ahead-of-print), p. 1
"100年前的这个月，美国数学月刊"《美国数学月刊》，印刷前，第1页
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Pub Date : 2023-11-06DOI: 10.1080/00029890.2023.2274242
Raymond Mortini
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Pub Date : 2023-10-20DOI: 10.1080/00029890.2023.2263299
Thomas F. Banchoff, Felix Günther
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
1827年，高斯证明了高斯曲率实际上是一个本征量，这意味着它可以通过表面内部的测量来计算。在此之前，曲面的曲率只能从外部计算，这意味着需要一个环境空间。高斯将这一非凡的发现命名为“惊人定理”。本文讨论了该定理在多面体曲面上的离散化形式。给出了离散高斯曲率的一般外在定义和内在定义是等价的一个初等证明。作者感谢两位匿名审稿人和编委会的宝贵意见。这项研究是在第二作者在波恩的马克斯普朗克数学研究所逗留期间开始的。该项目由德国研究基金会(Deutsche Forschungsgemeinschaft DFG)通过“几何和动力学离散化”合作研究中心TRR 109和德国卓越战略-柏林数学研究中心MATH+ (ec -2046/1，项目ID: 390685689)资助。作者简介:thomas F. BANCHOFF，布朗大学数学名誉教授，1967年至2015年任教于布朗大学。他于1964年在加州大学伯克利分校获得博士学位，并于1999年至2000年担任MAA主席。布朗大学数学系，邮编:151 Thayer Street, Providence RI 02912Thomas_Banchoff@brown.eduFelix g ntherfelix GÜNTHER, 2014年获得柏林理工大学Universität数学博士学位。在伊维特河畔布尔斯高等科学研究所、剑桥艾萨克·牛顿数学科学研究所、维也纳欧文Schrödinger国际数学与物理研究所、波恩马克斯·普朗克数学研究所和日内瓦大学担任博士后后，他于2018年回到Universität柏林工业大学。主要研究方向为离散微分几何和离散复分析。他还热衷于科学传播。柏林理工大学Universität，德国数学研究所8-3,Straße des 17。Juni 136, 10623柏林，Germanyfguenth@math.tu-berlin.de
{"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}
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.
{"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}
Pub Date : 2023-10-20DOI: 10.1080/00029890.2023.2263158
Timothy Sun
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.
{"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}
Pub Date : 2023-10-20DOI: 10.1080/00029890.2023.2266981
Beth Malmskog
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.
{"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}
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