How can a bicycle present as a unicycle? What happens if you keep iterating the logarithm of the absolute value? How can two mathematicians who were not alive at the same time share the credit for a formula? What is a good method for approximating the volume of a lake? Which functions are problematic for teaching vertical and horizontal stretches? What are hexagonal families and how are traits inherited among them? These captivating questions are among the many addressed in this issue of The College Mathematics Journal, along with five Proofs Without Words, a review of Frank Farris's handsome new book, the regular installments of Classroom Capsules, Problems & Solutions, and Media Highlights, and a poem on the 1867 meeting of two mathematical men named Charles. -*Brian Hopkins*

Vol 47 No 3, pp 160-239

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## ARTICLES

### How to Approximate the Volume of a Lake

Robert L. Foote and Han Nie

We study a nonlinear numerical integration method used by limnologists (scientists who study lakes) to approximate the volume of a lake. After proving an error estimate and making empirical comparisons, we suggest using Simpson's sum or integrated cubic splines.

To purchase from JSTOR: http://www.jstor.org/stable/10.4169/college.math.j.47.3.162

### Proof Without Words: Perfect Numbers and Triangular Numbers

Roger B. Nelsen

We show wordlessly that every even perfect number greater than six is one more than nine times a triangular number.

To purchase from JSTOR: http://www.jstor.org/stable/10.4169/college.math.j.47.3.171

### Iterating the Logarithmic Function

Xianglong Ni

We study the discrete dynamical system that results from iterating the absolute-valued logarithmic function, with different bases. Specifically, we analyze the periodic points, those that return to their original value after a number of iterations. By studying their paths under iteration, we determine how many exist for any given period length.

To purchase from JSTOR: http://www.jstor.org/stable/10.4169/college.math.j.47.3.172

### Proof Without Words: Square Triangular Numbers and Almost Isosceles Pythagorean Triples

Roger B. Nelsen

We illustrate wordlessly a one-to-one correspondence between square triangular numbers and almost isosceles Pythagorean triples.

To purchase from JSTOR: http://www.jstor.org/stable/10.4169/college.math.j.47.3.179

### Discrete and Smooth Bicycle “Unicycle” Paths

Amy Nesky and Clara Redwood

This paper explores the properties of bicycle paths in which the front wheel path and the back wheel path coincide. First, we extend previous work by establishing an increase in the number of points of inflection when the path is an infinitely smooth curve. Second, we consider a discrete model when the path consists of line segments and circular arcs; in this context, we prove conjectures on the complexity of these ``unicycle" paths including exponential growth of both the path length and total absolute curvature.

To purchase from JSTOR: http://www.jstor.org/stable/10.4169/college.math.j.47.3.180

### Proof Without Words: Nearly Cubic Pythagorean Boxes

Roger B. Nelsen

We show wordlessly that there are infinitely many nearly cubic Pythagorean boxes.

To purchase from JSTOR: http://www.jstor.org/stable/10.4169/college.math.j.47.3.190

### Proof Without Words: The Golden Ratio

Roger B. Nelsen

We employ a square with area 5 to determine the golden ratio (without using the quadratic formula).

To purchase from JSTOR: http://www.jstor.org/stable/10.4169/college.math.j.47.3.190

### What′s in a Name: Why Cauchy and Euler Share the Cauchy–Euler Equation

Adam E. Parker

The fact that that Cauchy was born after Euler's death should give pause as to why they are both identified with a well-known differential equation. We will examine the contributions of both these extremely proficient mathematicians. By studying the primary sources associated with them, we (re)discover some mathematics and also show Johann Bernoulli solved the equation years before Euler was born.

To purchase from JSTOR: http://www.jstor.org/stable/10.4169/college.math.j.47.3.191

### Proof Without Words: The Product-to-Sum Identities

John Molokach

We present a wordless proof of two basic trigonometric identities involving two angles, sine, and cosine.

To purchase from JSTOR: http://www.jstor.org/stable/10.4169/college.math.j.47.3.199

### Pedagogically Inconvenient Functions for Teaching Transformations

Todd Abel and Jeremy Brazas

Pedagogically inconvenient functions are those that look the same under two different types of transformations. We characterize those functions that look the same under vertical and horizontal stretches. In the continuous case, this leads to an expected conclusion; in the noncontinuous case, it does not. Though the question is quite simple and accessible, a full characterization leads us in some surprising mathematical directions.

To purchase from JSTOR: http://www.jstor.org/stable/10.4169/college.math.j.47.3.200

### Proof Without Words: Matchstick Triangles

Tom Edgar

We define the triangular matchstick numbers and wordlessly prove that each one is three times a triangular number as well as the difference of two triangular numbers.

To purchase from JSTOR: http://www.jstor.org/stable/10.4169/college.math.j.47.3.207

### Inheritance Relations of Hexagons and Ellipses

Mahesh Agarwal and Narasimhamurthi Natarajan

After establishing a notion of parent and child hexagons, we examine the ways in which some properties are hereditary. In particular, we use results of Pascal and Brianchon on circumscribing and inscribing ellipses, respectively, to show that these are similar to recessive traits in that they can skip generations. We also describe how to construct hexagons that have both circumscribing and inscribing ellipses.

To purchase from JSTOR: http://www.jstor.org/stable/10.4169/college.math.j.47.3.208

### Babbage and Carroll in the Silent Workshop, 1867

Neil Aitken

To purchase from JSTOR: http://www.jstor.org/stable/10.4169/college.math.j.47.3.215

## Classroom Capsules

### A Visual Approach to Geometric Series with Negative Ratio

Amal Sharif-Rasslan

An earlier visual approach for summing an infinite geometric series with positive ratios uses a book with some pages left in hand and some pages kept at each stage. We extend this book demonstration to geometric series with negative ratios by also throwing away some pages at each stage.

To purchase from JSTOR: http://www.jstor.org/stable/10.4169/college.math.j.47.3.216

### The Chu-Vandermonde Identity via Leibniz′s Identity for Derivatives

Michael Spivey

We give a proof of the Chu–Vandermonde identity using only Leibniz′s identity for derivatives.

To purchase from JSTOR: http://www.jstor.org/stable/10.4169/college.math.j.47.3.219

## Problems and Solutions

Problems and Solutions: 221-227

To purchase from JSTOR: http://www.jstor.org/stable/10.4169/college.math.j.47.3.221

## Book Review

*Creating Symmetry: The Artful Mathematics of Wallpaper* By Frank A. Farris

Reviewed by: Heidi Burgiel

To purchase from JSTOR: http://www.jstor.org/stable/10.4169/college.math.j.47.3.228

## Media Highlights

To purchase from JSTOR: http://www.jstor.org/stable/10.4169/college.math.j.47.3.232