The lead article of Issue 2 is a survey of optimal line packing, leading readers through equiangular tight frames derived from some (but not all) Platonic solids to conjectures and open questions with a blog link for the latest developments in this active field devoted to overcomplete systems. Anticipating the 13th Gathering 4 Gardner next month, Art Benjamin shares some things he learned from John H. Conway at an earlier Gathering—you may be surprised how effectively you can use your fingers to find prime factors of some four digit numbers. (By the way, Conway also appears in the extensive bibliography of the frames article.) And fans of 1, 1, 2, 3, 5, 8, 13 will enjoy a Proof Without Words developed for a middle school classroom to show a result about squares of consecutive Fibonacci numbers without citing induction.

Vol. 49, No. 2, pp. 81-160

##### JOURNAL SUBSCRIBERS AND MAA MEMBERS:

To read the full articles, please log in to the member portal by clicking on 'Login' in the upper right corner. Once logged in, click on 'My Profile' in the upper right corner.

## ARTICLES

### A Short Introduction to Optimal Line Packings

p. 82.

Dustin G. Mixon & James Solazzo

How do you arrange lines in Euclidean space so that the smallest interior angle is as large as possible? This article provides an illustrated survey of recent work in this problem, with an eye on opportunities for undergraduate research.

DOI: 10.1080/07468342.2018.1421364

### Proof Without Words: Volume of a Pedestal Prismoid

p. 92.

Lucas Amiras

We wordlessly determine the volume of a special rectangular prismoid by decomposition into solids and their transformations into certain parallelepipeds.

DOI: 10.1080/07468342.2018.1424423

### Why the Centroid is the Centroid: Modern Variations on a Theme of Archimedes

p. 93.

William C. Mercier

Iteratively subdividing a triangle, we locate its center of mass in a manner that melds two approaches taken by Archimedes. Still keeping well within his formalism, we do the same again using an iterative method employed to produce fractals. Finally, again using this second method, we locate the centroid of the tetrahedron to a modern standard of rigor.

DOI: 10.1080/07468342.2018.1420982

### A Fourth Century Theorem for Twenty-First Century Calculus

p. 103.

Andrew Leahy

The centroid theorems of Pappus (or the Pappus–Guldin theorems, or the Guldin theorems) show deep connections between areas, arc lengths, volumes of revolution, surfaces of revolution, and centers of gravity. Because these theorems are historically important results that easily extend and imply some of the most well-known formulas in integral calculus, we argue that they should be given a more central role in calculus instruction.

DOI: 10.1080/07468342.2018.1421817

### Newton’s Shell Theorem via Archimedes’s Hat Box and Single-Variable Calculus

p. 109.

Peter McGrath

Newton’s shell theorem asserts that the net gravitational force between a point particle and a sphere with uniform mass density is the same as the force in the situation where the sphere is replaced by a point particle at its center with the same total mass. We give an exposition of this theorem using only tools from introductory one-variable calculus. A key simplification is a result of Archimedes that the area of the region on a sphere between two parallel planes depends only on the separation between the planes, not on their position relative to the sphere.

DOI: 10.1080/07468342.2018.1411655

### Cutting Against the Grain: Volumes of Solids of Revolution via Cross-Sections Parallel to the Rotation Axis

p. 114.

Kevin P. Knudson

Rather than slicing solids of revolution perpendicular to the axis of rotation, we consider what happens when we cut the surface with planes parallel to the axis. Using this approach to explore the solid known as Gabriel's horn, which has finite volume and infinite surface area, yields insights into the growth rate of the logarithm function.

DOI: 10.1080/07468342.2018.1411656

### Proof Without Words: Sum of Squares of Consecutive Fibonacci Numbers

p. 121.

Tim Price

We present a visual proof that the sum of the squares of two consecutive Fibonacci numbers is also a Fibonacci number.

DOI: 10.1080/07468342.2018.1424425

### Factoring Numbers with Conway’s 150 Method

p. 122.

Arthur T. Benjamin

We describe a “handy” method, due to John Conway, for quickly finding all relatively small prime factors of 3-digit and 4-digit numbers.

DOI: 10.1080/07468342.2017.1411141

### Euler's Sine Product Formula: An Elementary Proof

p. 126.

David Salwinski

We provide a proof of Euler's sine product formula using simple techniques of calculus and illustrate how our method can be used to prove similar product formulas for cosine as well as hyperbolic sine and hyperbolic cosine. We also derive bounds on the partial products and explore some consequences of these formulas.

DOI: 10.1080/07468342.2018.1419703

## Classroom Capsules

### The Double-Sidedness of Matrix Inverses; Yet Another Proof

p. 136.

Esther M. García-Caballero & Samuel G. Moren

We prove the two-sidedness of matrix inverses using only a basic fact about when spanning vectors of a finite dimension space form a basis.

DOI: 10.1080/07468342.2018.1426315

### Can a Subset's Topology Detect Continuous Extensions?

p. 138.

Mike Krebs

Building on a common exercise in point-set topology, we establish three lemmas that allow us to answer the following question. Is there a topology on the rational numbers where a function is continuous if and only if the function extends to a continuous function on the reals?

DOI: 10.1080/07468342.2017.1397450

## Problems and Solutions

p. 140.

DOI: 10.1080/07468342.2018.1432203

## Book Review

*Combinatorics: A Very Short Introduction* by Robin Wilson

and *Infinity: A Very Short Introduction* by Ian Stewart

p. 147.

Reviewed by Brian Hopkins

DOI: 10.1080/07468342.2018.1432202

## Media Highlights

p. 153.

DOI: 10.1080/07468342.2017.1419010