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College Mathematics Journal Contents—May 2015

This issue of The College Mathematics Journal has something curved, spherical, or spinning in every article. David Richeson (winner of the 2010 Euler Book Prize) leads by speculating who first proved that the diameter of a circle over its circumference is a constant. While the poi spinning diagrams of Eleanor Farrington's article may not be as complex as Cholasinth Chorsakul's dizzying cover, they bring an invigorating visual and physical component to parametric equations. Boualem and Brouzet survey many aspects of what it means to be a circle in addressing a problem from a French high school, while Foote and Sun venture into other geometries and Peters goes up in dimension. The Classroom Capsules, Problems & Solutions, and Media Highlights span the journal's normal breadth of mathematical topics, and Tina Garrett's review lauds Richard Stanley's new book devoted to the Catalan numbers. -Brian Hopkins

Vol 46 No 3, pp 161-240


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Circular Reasoning: Who First Proved That C Divided by d Is a Constant?

David Richeson

We argue that Archimedes proved that the ratio of the circumference of a circle to its diameter is a constant independent of the circle and that the circumference constant equals the area constant. He stated neither result explicitly (in surviving material), but both are implied by his work. His proof required the addition of two axioms beyond those in Euclid’s Elements.

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Proof Without Words: Each But Two Triangular Numbers Is a Sum of Three Triangular Numbers

Roger B. Nelsen

We show wordlessly that each triangular number except 1 and 6 is a sum of three triangular numbers.

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Parametric Equations at the Circus: Trochoids and Poi Flowers

Eleanor Farrington

Poi spinning is a performance art, related to juggling, where weights on the ends of short chains are swung to make interesting patterns. We study a certain class of moves for poi where the patterns created are centered trochoids. Like all curves in the cycloid family, they are best expressed using parametric equations. Using the calculus of the curves, we find that there are just a few places where one pattern can be smoothly transformed into another.

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On the Shrinking Volume of the Hypersphere

Michael H. Peters

As you increase the dimension of a sphere with fixed radius into the realm of hypergeometry, the volume increases—for a while. We explore the counterintuitive result that, as we keep adding dimensions, eventually the volume decreases.

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Partial Proof Without Words: Shaping Some Cases of the Erdős–Straus Conjecture

Juan D. Serna

We provide visual proofs for three infinite cases of the Erdős–Straus conjecture on Egyptian fractions.

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An Intrinsic Formula for the Cross Ratio in Spherical and Hyperbolic Geometries

Robert L. Foote

The cross ratio is a well known and useful invariant in Euclidean and projective geometries. Since it is invariant under Möbius transformations, it extends to spherical and hyperbolic geometries. It seems less useful in these settings, however, due to the lack of an intrinsic interpretation. We give such an interpretation and use it to extend the formula of Apollonius for a circle to constant curvature curves in these geometries.

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Rational and Implicit Equations for Some Polar Curves

Dave Boyles

Given an implicit equation for an algebraic plane curve, we consider when and how it is possible to find rational parametric functions that also draw the curve. Using sequences similar to the Chebyshev polynomials, we detail examples, including Maclaurin sectrices, Ceva sectrices, and polar flowers.

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To Be (a Circle) or Not to Be?

Hassan Boualem and Robert Brouzet

We begin with a high school exercise, determining whether a particular equation describes an arc of a circle. We give several proofs that it does not, exploring many properties of circles along the way. Yet the curve is the arc of a conic.We then explore which equations of a generalized family are arcs of conics and give some properties of the resulting Lamé curves.

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On the Inverse Curvature Problem

Adam Glesser, James Shade and Bogdan D. Suceavă

We pose and investigate the inverse curvature problem. That is, we explore when elements of a certain family of monomials represent the curvature function of a curve specified by combinations of elementary functions. The main result utilizes an integration theorem of Chebyshev.

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Classroom Capsules

An Even Simpler Proof of the Right-Hand Rule

Eric Thurschwell

We present a trigonometric proof of the right-hand rule.

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An Inductive Proof of the Compactness of the Closed Unit Ball of an Arbitrary Dimension

Haryono Tandra

Using the mathematical induction, we present a proof of the compactness of an arbitrary dimension closed unit ball that generalizes the standard proof of the compactness of an interval.

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Problems and Solutions

Problems and Solutions: 220-227

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Book Review

Catalan Numbers by Richard P. Stanley

Reviewed by Kristina C. Garrett

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Media Highlights

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