February is the month for math lovers… In this issue of the *Monthly*, Annalisa Crannell, Marc Frantz, and Fumiko Futamura help us understand squares and their perspective images through projective geometry and art; Allen Stenger demonstrates how to use experimental mathematics to aid in solving *Monthly* problems; Michael Maltenfort discusses additive systems of sets of natural numbers (most helpful when making change), and Bisharah Libbus, Gordon Simons, and Yi-Ching Yao consider how to independently rotate finite sets of points in *n*–dimensions so as to bring them all close together, a problem with applications to rigid-body kinematics, robotics, and biology.

In the Notes section, you can find infinite products for expressions involving powers of *e*, an elementary proof of when sums of quadratic residues equal sums of nonresidues, conditions for when an average of roots of unity is an algebraic integer, and which “small” equilateral polygons have maximum area.

As always, there is a selection of problems to temp you; perhaps you might use methods of experimental math to solve them. Finally, in this month’s review, Alan Baker discusses the philosophies of mathematics and recommends several books for your consideration.

— *Susan Jane Colley, Editor*

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## Table of Contents

### The Image of a Square

p. 99.

Annalisa Crannell, Marc Frantz, and Fumiko Futamura

*Every quadrangle is the perspective image of a square.* We illustrate this statement by using perspective art techniques and by analogy to the visualization of conic sections. We also give examples of how understanding perspective images of squares can be applied fruitfully in the areas of photogrammetry (determining true relative sizes of real-world objects from a photograph) and linear algebra (more specifically, in the decomposition of projective transformations).

DOI: http://dx.doi.org/10.4169/amer.math.monthly.124.2.99

### Experimental Math for Math Monthly Problems

p. 116.

Allen Stenger

Experimental mathematics is a newly developed approach to discovering mathematical truths through the use of computers. In this paper, we look at how these techniques can be applied to help solve six problems that have appeared in the Problems section of the *Monthly*. The paper has examples of constant recognition, sequence recognition, and integer relation detection.

DOI: http://dx.doi.org/10.4169/amer.math.monthly.124.2.116

### Characterizing Additive Systems

p. 132.

Michael Maltenfort

An additive system is a collection of sets that gives a unique way to represent either all nonnegative integers, or all nonnegative integers up to some maximum. A structure theorem of de Bruijn gives a certain form for an additive system of infinite size. This form is not, in general, unique. We improve de Bruijn’s theorem by finding a unique form for an additive system of arbitrary size. Our proof gives a concrete construction that allows us to test easily whether a collection of sets is an additive system. We also show how to determine most of the structure of an additive system if we are only given its union.

DOI: http://dx.doi.org/10.4169/amer.math.monthly.124.2.132

### Rotating Multiple Sets of Labeled Points to Bring Them Into Close Coincidence: A Generalized Wahba Problem

p. 149.

Bisharah Libbus, Gordon Simons, and Yi-Ching Yao

While attempting to better understand the 3-dimensional structure of the mammalian nucleus as well as a rigid-body kinematics application, the authors encountered a naturally arising generalized version of the Wahba (1965) problem concerned with bringing multiple sets of labeled points into close coincidence after making appropriate rotations of these sets of labeled points. Our solution to this generalized problem entails the development of a computer algorithm, described and analyzed herein, that generalizes and utilizes an analytic formula, derived by Grace Wahba (1965), for determining space satellite attitudes, that task being to find a suitable rotation that brings one set of *m* labeled points into close coincidence, in a least-squares sense, with a second set of *m* labeled points.

DOI: http://dx.doi.org/10.4169/amer.math.monthly.124.2.149

## Notes

### Generalized Infinite Products for Powers of *e*^{1/k}

p. 161.

Scott Ginebaugh

Catalan and Pippenger discovered infinite products for various values related to *e*. Based on these, infinite products for *e*^{1/2} and *e*^{2/}3 were found by Sondow and Yi, who furthermore conjectured that the products could be generalized for a power of *e*^{1/k} . The discoveries in this paper result from attempting to prove this conjecture. By generalizing Sondow and Yi’s products, infinite products for powers of *e*^{1/k} are found.

DOI: http://dx.doi.org/10.4169/amer.math.monthly.124.2.161

### Sums of Quadratic Residues and Nonresidues

p. 166.

Christian Aebi and Grant Cairns

It is well known that when a prime *p* is congruent to 1 modulo 4, the sum of the quadratic residues equals the sum of the quadratic nonresidues. In this note, we give elementary proofs of V.-A. Lebesgue’s analogous results for the case where *p* is congruent to 3 modulo 4.

DOI: http://dx.doi.org/10.4169/amer.math.monthly.124.2.166

### A Note on Average Roots of Unity

p. 170.

Chatchawan Panraksa and Pornrat Ruengrot

We consider the problem of characterizing all functions* f* defined on the set of integers modulo *n* with the property that an average of some *n*th roots of unity determined by *f* is always an algebraic integer. Examples of such functions with this property are linear functions. We show that, when *n* is a prime number, the converse also holds. That is, any function with this property is representable by a linear polynomial. Finally, we give an application of the main result to the problem of determining self-perfect isometries for the cyclic group of prime order *p*.

DOI: http://dx.doi.org/10.4169/amer.math.monthly.124.2.170

### Maximal Area of Equilateral Small Polygons

p. 175.

Charles Audet

We show that among all equilateral polygons with a given number of sides and the same diameter, the regular polygon has the maximal area.

DOI: http://dx.doi.org/10.4169/amer.math.monthly.124.2.175

## Problems and Solutions

p. 179.

DOI: http://dx.doi.org/10.4169/amer.math.monthly.124.2.179

## Book Review

p. 188.

### The Philosophies of Mathematics

Alan Baker

DOI: http://dx.doi.org/10.4169/amer.math.monthly.124.2.188

## MathBits

### 100 Years Ago This Month in *The American Mathematical Monthly*

p. 148.

### Another Proof That There Are Infinitely Many Primes

p. 169.

DOI: http://dx.doi.org/10.4169/amer.math.monthly.124.2.169