Mathematics Magazine - December 2017

Need something to read over the holidays? This issue of the Magazine has plenty to offer. Articles introduce a new game based on knot theory, generalize Viviani’s theorem, represent the natural logarithm by an infinite series to highlight the logarithm’s concavity, and recover the binomial theorem as the unique solution of an nth order linear nonhomogeneous ordinary differential equation.

Other articles consider limits of indeterminate forms without L’Hospital’s rule using the squeeze theorem, revisit Laplace’s rule of succession and its use in Colley’s method via integration by parts, induction, and probability, and find the osculating circle (the circle used to approximate a curve at a point with the same curvature as the curve) without the normal vector. There is even a two-pack of Euler-related articles, one about Euler and the prime number theorem and another on a dissection proof of a series by Euler.

There is an art interview with Chris K. Palmer, who works in origami, a crossword puzzle about the Joint Mathematics Meetings, and a Pinemi puzzle. There are reviews to read in the Reviews section and problems and solutions to check out in the Problems section.

—Michael A. Jones, Editor

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Vol. 90, No. 5, pp. 321 – 400

Articles

Letter from the Editor

p. 322.

The Region Unknotting Game

p. 323.

Sarah Brown, Francisco Cabrera, Riley Evans, Gianni Gibbs, Allison Henrich, and James Kreinbihl

Motivated by recent work by A. Shimizu on a newly discovered unknotting operation and inspired by previous work on knot games, we introduce the Region Unknotting Game. We play the game on several types of knot diagrams, developing both our spatial intuition and our understanding of the structure of knots.

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Loci of Points Inspired by Viviani's Theorem

p. 338.

Elias Abboud

We consider loci of points such that their sum of distances or sum of squared distances to each of the sides of a given triangle is constant. These loci are inspired by Viviani’s theorem and its extension. The former locus is a line segment or the whole triangle and the latter locus is an ellipse.

To purchase from JSTOR: 10.4169/math.mag.90.5.338

The Osculating Circle Without the Unit Normal Vector

p. 347.

Hossein Hosseini Giv

The center of an osculating circle of a plane curve is usually found via the unit normal vector. In this paper we show that, once the radius of this circle is given, the center can be found using the calculus of parametric plane curves. This approach may be instructive for students.

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An Infinite Series that Displays the Concavity of the Natural Logarithm

p. 353.

David M. Bradley

The natural logarithm can be represented by an infinite series which, in contrast with its Taylor series, converges for all positive real values of the variable, and makes the fundamental property of concavity patently obvious. In principle, the series can be used to calculate the natural logarithm at any point of its domain using only the operations of addition, subtraction, multiplication, and square root extraction repeatedly.

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Could Euler Have Conjectured the Prime Number Theorem?

p. 355.

Simon Rubinstein-Salzedo

In this article, we investigate how Euler might have been led to conjecture the prime number theorem, based on what he knew. We also speculate on why he did not do so.

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A Dissection Proof of Euler's Series for 1 — γ

p. 360.

Mits Kobayashin

We demonstrate a visualization of Euler’s series for 1 − γ, where γ is the Euler–Mascheroni constant.

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Colley's Coin: Ranking Sports Teams With Laplace's Rule of Succession

p. 365.

Carl V. Lutzer

The Colley ranking method was adopted by the NCAA to rank college football teams and has also been used by ecologists and social scientists. This ranking method is based on the ratio (wins+1)/(games played+2). At first glance, this ratio appears close enough to the familiar (wins)/(games played) ratio that many people feel it should be easily understood, and yet most people find it strikingly unintuitive. In this sense, the ratio is in a kind of mathematical “zombie zone”—simple enough that it should be intuitive and yet not intuitive. In this note, we provide an elementary derivation of the ratio that is accessible to undergraduates and ties together three disparate parts of the curriculum: integration by parts, mathematical induction, and probability.

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On Indeterminate Forms of Exponential Type

p. 371.

Jinsen Xiao and Jianxun He

We apply the squeeze theorem, instead of L’Hospital’s rule, to evaluate limits in indeterminate form of exponential type. We do not require differentiability, instead needing the boundedness of a quotient.

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The Binomial Theorem Procured From the Solution of an ODE

p. 375.

Kuldeep Kumar Kataria

We obtain the binomial theorem as a unique solution of an nth order linear nonhomogeneous ordinary differential equation with constant coefficients and given initial conditions.

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Joint Mathematics Meetings 2018

p. 378.

Brendan Sullivan

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Chris K. Palmer: Origami in Action

p. 380.

Amy L. Reimann and David A. Reimann

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Problems and Solutions

p. 383.

Proposals, 2031-2035

Quickies, 1075-1076

Solutions, 2001-2010

Answers, A1075-A1076

To purchase from JSTOR: 10.4169/math.mag.90.5.383

Reviews

p. 395.

Babylonian trig; crowd-sourcing Euclid; does math enhance reasoning?; gerrymandering

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Referee Thank You

p. 397.

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