Pub Date : 2023-01-01DOI: 10.1080/0025570x.2023.2165864
Brian Hopkins, R. Nelsen
Summary We give rectangular tiling proofs for sum-product identities are recurrences involving the Padovan and Perrin numbers, closely related third order recursively-defined integer sequences.
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Pub Date : 2023-01-01DOI: 10.1080/0025570x.2023.2169513
J. Rosenhouse
Welcome to the inaugural issue of Mathematics Magazine for 2023! We have another bumper crop of expository excellence for your reading pleasure. Our lead article is a survey of closed hypocycloids and epicycloids by Zarema Seidametova and Valerii Temnenko. Readers are probably familiar with the cycloid, which is the curve traced out by a point on the circumference of a circle as it rolls along a line. If instead we have the circle roll around the inside of a second circle, the result is a hypocycloid, and if it rolls around the outside of a second circle we get an epicycloid. The resulting shapes are some of the most beautiful and elegant in all of mathematics. In addition to providing us with our cover images for this issue, Seidametova and Temnenko suggest an insightful classification scheme for these curves. José Cereceda takes his inspiration from Nicomachus’ identity. You know the one I mean: Summing the first n numbers and squaring is the same as summing the first n cubes. Cereceda guides us through the fascinating world of arithmetic hypersums to prove a generalization of this theorem. Sums also feature prominently in Russell Gordon’s contribution. Every calculus student knows the standard convergence tests for infinite series, but Raabe’s test is rarely included in the syllabus. Gordon makes a convincing case that this omission is unfortunate. He shows how to use Raabe’s test to prove the convergence of various series that defy the standard tests. He also shows how some ingenuity can be used to evaluate sums that at first blush seem hopelessly intractable. Evin Liang rounds out the longer articles for this issue by returning us to Triphos— “a world without subtraction.” Triphos was last explored in this Magazine in our October 2019 issue. The authors of that previous article posed a variety of questions about the geometry and trigonometry of this strange world. Liang accepted the challenge, with the results presented in his wonderfully lucid article. The shorter pieces also provide much food for thought. Greg Dresden explores connections among the Fibonacci numbers and Chebyshev polynomials. Raymond Mortini and Peter Pflug prove that a strip, meaning a region bounded by two parallel lines, is the only open convex set that disconnects the plane. This is one of those things that seems obvious until you try to prove it. Ricardo Podestá takes an elegant, visual approach to proving that square roots are irrational. Tom Edgar explores the standard means—arithmetic, geometric, harmonic, and quadratic. He takes a clever, physics-based approach to proving the familiar inequalities among them. Frédéric Paul contributes an insightful discussion of the relationships between two famous analytic inequalities due to Maclaurin and Bernoulli. And Quang Hung Tran rounds out the proceedings by using Ptolemy’s theorem on cyclic quadrilaterals to prove a generalization of the Pythagorean theorem. We also have problems, reviews, proofs without words, and the p
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Pub Date : 2023-01-01DOI: 10.1080/0025570X.2023.2167397
Zarema S. Seidametova, V. Temnenko
Summary The paper describes a classification of closed epicycloids and hypocycloids into three classes: “odd/odd,” “even/odd,” “odd/even.” A subset of “perfect” epicycloids and hypocycloids that do not have self-intersection points has been identified. A new composite geometric object is constructed: the Euler Ring of Rings, consisting of a perfect epicycloid and a perfect hypocycloid with the same indices.
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Pub Date : 2023-01-01DOI: 10.1080/0025570X.2023.2166328
Q. H. Tran
Summary We establish a generalization of the Pythagorean theorem with a proof using Ptolemy’s theorem.
摘要利用托勒密定理的一个证明，建立了勾股定理的推广。
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Pub Date : 2023-01-01DOI: 10.1080/0025570X.2023.2165855
R. Mortini, P. Pflug
Summary We show that the number of complementary components of an open convex set in the plane is 0, 1, or 2 by showing that isometric images of strips (where I is a bounded open interval) are the only open convex sets in with disconnected complement.
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Pub Date : 2023-01-01DOI: 10.1080/0025570X.2023.2165861
J. Cereceda
Summary We provide a new proof of a simple generalization of the famous identity by making use of the hyper-sums of powers of integers.
利用整数幂的超和，给出了著名恒等式的一个简单推广的新证明。
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Pub Date : 2023-01-01DOI: 10.1080/0025570X.2023.2167431
Tom Edgar
Summary We provide two visual proofs of the two-variable harmonic mean-geometric mean-arithmetic mean-quadratic mean inequalities: one using a center of mass model and one using moments of mass.
{"title":"Balanced and Unbalanced: Physical Proofs of the Mean Inequalities","authors":"Tom Edgar","doi":"10.1080/0025570X.2023.2167431","DOIUrl":"https://doi.org/10.1080/0025570X.2023.2167431","url":null,"abstract":"Summary We provide two visual proofs of the two-variable harmonic mean-geometric mean-arithmetic mean-quadratic mean inequalities: one using a center of mass model and one using moments of mass.","PeriodicalId":18344,"journal":{"name":"Mathematics Magazine","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45014378","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-01-01DOI: 10.1080/0025570X.2023.2168436
R. Podestá
Summary We give a geometric proof that is irrational for n = 3, 5, 7 by adapting Tennenbaum’s geometric proof that is irrational. We also show that this method cannot be used to prove the irrationality of for a bigger n.
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Pub Date : 2023-01-01DOI: 10.1080/0025570X.2023.2167394
B. Bajnok
Summary We present the problems and solutions to the 51st Annual United States of America Mathematical Olympiad.
我们提出了第51届美国年度奥林匹克数学竞赛的问题和解决方案。
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Pub Date : 2023-01-01DOI: 10.1080/0025570X.2023.2166332
F. Paul
Summary We revisit the article “Maclaurin’s inequality and a generalized Bernoulli inequality” published in this Magazine in 2014, by presenting a more elementary link between Maclaurin’s inequality and Bernoulli’s inequality.
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