Letter From The Editor

Abstract

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 problems and solutions from the 51st annual USA Mathematical Olympiad. That should keep you busy until we do this all again in our April issue.