The Pythagorean Theorem is woven though the May issue of *The College Mathematics Journal*. Nelsen and Ren provide three Proofs Without Words extending the famed theorem in various ways while a composite effort of Stankewicz and Coll, Davis, Hall, Magnant, and Wang explore its use in developing "nice" box optimization problems in calculus. Crisman and Veatch's "Reinventing Heron" uses a calculus approach to prove this triangle result equivalent to the Pythagorean Theorem. Calculus also makes an unexpected appearance in Clancy and Kifowit's analysis of Bobo's integer sequence. Works of Dürer and Mantegna, along with cover art of Harry Brodsky, support Dillon's investigation of projective geometry in the classroom. And that describes not even half of the issue.—*Brian Hopkins*

Vol. 45, No. 3, pp. 162-239.

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## ARTICLES

### Locating Unimodular Roots

Michael A. Brilleslyper and Lisbeth E. Schaubroeck

We investigate a simple family of trinomials and characterize the number and location of all their unimodular roots in terms of a divisibility condition on the sum of the exponents. The main result depends on a classical theorem about Diophantine equations.

To purchase from JSTOR: http://dx.doi.org/10.4169/college.math.j.45.3.162

### Projective Geometry for All

Meighan Dillon

This article treats projective geometry as arising from the perspective problem addressed by Renaissance artists while sketching a program for teaching the material to students at all levels.

To purchase from JSTOR: http://dx.doi.org/10.4169/college.math.j.45.3.169

### Proof Without Words: Pythagorean Quadruples

Roger Nelsen

We visually prove a formula that generates Pythagorean quadruples.

To purchase from JSTOR: http://dx.doi.org/10.4169/college.math.j.45.3.179

### Integer Solutions to Box Optimization Problems

Vincent Coll, Jeremy Davis, Martin Hall, Colton Magnant, James Stankewicz, Hua Wang

Two approaches contribute to the problem of finding integer side-lengths for open rectangular and polygonal boxes that optimize volume. One method adapts Pythagorean triples to an ellipse, while the other is based on counting factorizations.

To purchase from JSTOR: http://dx.doi.org/10.4169/college.math.j.45.3.180

### Proof Without Words: Sine Sum Identity

Long Wang

We give a visual proof of the sine sum identity.

To purchase from JSTOR: http://dx.doi.org/10.4169/college.math.j.45.3.190

### Reinventing Heron

Karl-Dieter Crisman and Michael H. Veatch

Heron’s formula gives the area of a triangle from the side lengths alone. Normally, this theorem is motivated by geometry or trigonometry. But how might calculus-style approximation lead us to such a formula? Here, we examine this idea, as well as the potential accuracy of such approximations.

To purchase from JSTOR: http://dx.doi.org/10.4169/college.math.j.45.3.191

### Proof Without Words: Pythagoras for a Right Trapezoid

Guanshen Ren

We wordlessly prove an identity for right trapezoids using the Pythagorean theorem.

To purchase from JSTOR: http://dx.doi.org/10.4169/college.math.j.45.3.198

### A Closer Look at Bobo’s Sequence

Daniel T. Clancy and Steven J. Kifowit

Any sum of reciprocals of consecutive natural numbers must eventually exceed 1. The final term of such a sum is a function of the initial term. In a 1995 Classroom Capsule, E. Ray Bobo described some properties of that function and posed several questions regarding its possible values. We answer some of those questions, primarily using integral approximations from calculus.

To purchase from JSTOR: http://dx.doi.org/10.4169/college.math.j.45.3.199

### The Rank of Recurrence Matrices

Christopher Lee and Valerie Peterson

A recurrence matrix is defined as a matrix whose entries (read left-to-right, row-by-row) are sequential elements generated by a linear recurrence relation. The maximal rank of this matrix is determined by the order of the corresponding recurrence. In the case of an order-two recurrence, the associated matrix fails to have full rank whenever the ratio of the two initial values of the sequence is an eigenvalue of the relation.

To purchase from JSTOR: http://dx.doi.org/10.4169/college.math.j.45.3.207

### Proof Without Words: Pythagoras for a Clipped Rectangle

Guanshen Ren

We wordlessly prove an identity for clipped rectangles using the Pythagorean theorem.

To purchase from JSTOR: http://dx.doi.org/10.4169/college.math.j.45.3.216

## CLASSROOM CAPSULES

### Johnson’s Three Circles Theorem Revisited

O. A. S. Karamzadeh

A new elementary proof of Johnson’s Theorem follows from a result on three arbitrary circles that meet in a point.

To purchase from JSTOR: http://dx.doi.org/10.4169/college.math.j.45.3.217

### Computing Volumes of Solids of Revolution with Double Integrals

Jorge Martín-Morales and Antonio M. Oller-Marcén

We show that the disk and shell methods for computing volume are connected by Fubini’s theorem. The same double integral allows us to derive Pappus’ volume theorem.

To purchase from JSTOR: http://dx.doi.org/10.4169/college.math.j.45.3.219

## PROBLEMS AND SOLUTIONS

Problems: Correction to 1018, 1026-1030

Solutions: 1001-1005

To purchase from JSTOR: http://dx.doi.org/10.4169/college.math.j.45.3.222

## REVIEW

*Love and Math: The Heart of Hidden Reality* by Edward Frenkel

reviewed by Tanya Khovanova

To purchase from JSTOR: http://dx.doi.org/10.4169/college.math.j.45.3.230

## MEDIA HIGHLIGHTS

To purchase from JSTOR: http://dx.doi.org/10.4169/college.math.j.45.3.232