# Mathematics Magazine - June 2005

### ARTICLES

In the Shadow of Giants: A Section of American Mathematicians, 1925-1950
David E. Zitarelli
175-191
The history of mathematics generally studies the work of giants in the field, like Newton, Euler, Gauss, Riemann, Poincaré, and Hilbert. When restricted to America, the towering figures include E. H. Moore, Veblen, Birkhoff, R. L. Moore, and Stone. But what about the professional activities of rank-and-file mathematicians? This article answers this question by examining the lives and work of five mathematicians who coalesced during 1925-50 to found and establish an MAA section. A closer examination reveals a surprisingly impressive array of outstanding achievements from individuals who deserve to be rescued from their present obscurity.

Cartoon
In De Morgan’s Grocery: The checker asks, Â“Not (not paper and not plastic)?Â”
Peter Schumer, with art by Amy Schumer and Eddingel Recaido
191

Farmer Ted Goes 3D
Shawn Alspaugh
192-204
One of the classic problems every calculus student faces involves minimizing the perimeter of rectangles with a fixed area. The answer often involves noninteger side lengths for the resulting square. In 1999, Greg Martin studied a variation of this problem, requiring all side lengths to be integers. We expand Martin's work into the third dimension. We define a sequence of numbers that are the volumes of rectangular boxes whose volume to surface area ratio is smallest among all such boxes with integer side lengths and smaller or equal volume. We then characterize these numbers, noticing they form three distinct groups that can be studied more carefully. We also discuss difficulties that arise when dealing with three dimensions as opposed to two, and give a theorem for determining if a given volume will minimize the volume to surface area ratio.

Pythagorean Triples and Inner Products
Larry J. Gerstein
205-213
The lattice generated by a set of linearly independent vectors is the set of linear combinations of those vectors having integer coefficients. In the present paper we use elementary properties of lattices to give a new perspective on Pythagorean triples, a topic in number theory that has been of interest for millennia. In the classical setting, formulas for the production of Pythagorean triples were derived from arithmetic manipulations in combination with elementary properties of congruences. But here the triples are shown to correspond to so called "isotropic" lattice vectors in a suitable three dimensional inner product space. Along the way we recapture a 19th-century theorem of Leonard Eugene Dickson, and we briefly discuss Pythagorean triples over rings other than the integers.

Proof Without Words: Viviani’s Theorem
Ken-ichiroh Kawasaki
213
In an equilateral triangle, the sum of the distances from any interior point to the three sides is equal to the altitude of the triangle. This theorem is proved by rotating certain equilateral triangles constructed inside the original triangle.

### NOTES

Wafer in a Box
Ralph Alexander and John E. Wetzel
214-220
How large a wafer can fit in a 5 by 7 by 8 box? How about a 5 by 7 by 9 box? We find an explicit formula for the radius of the largest disk that fits in an a by b by c rectangular box.

A Theorem of Frobenius and Its Applications
Dinesh Khurana and Anjana Khurana
220-225
The following fundamental theorem was proved by Frobenius more than hundred years back in 1895:

If d is a divisor of the order of a finite group G, then the number of solutions of xd=1 in G is a multiple of d.

The result has many applications in Group theory and Number theory. We present the result and some of its applications in a way that uses only elementary knowledge of Group theory.

Proof Without Words: (0,1) and [0,1] Have the Same Cardinality
Kevin Hughes and Todd K. Pelletier
226

A Â“BaseÂ” Count of the Rationals
Brian D. Ginsberg
227-228
Cantor’s classic proof that the rationals are countable is presented in every set theory course in the country, but this is not the only way to prove that result. In this Note, we exhibit a concrete numerical method for counting the rationals that also works to count the algebraic numbers and other sets with similar desirable properties.

Covering Systems of Congruences
J. Fabrykowski and T. Smotzer
228-231
The note considers problems related to covering the set of integers by the finite unions of arithmetic progressions. There are two well-known conjectures on the topic: the Selfridge conjecture which states that the set of integers cannot be covered by the finite union of arithmetic progressions having all differences odd and distinct, and the Schinzel conjecture that states if the set of integers is covered by the finite union of arithmetic progressions, then there exist two differences such that one is a divisor of another. It has been known that the Selfridge conjecture implies the Schinzel conjecture, but the proof of this fact is not elementary and is based on the theory of irreducibility of polynomials. The authors provide a simple and elementary proof of this assertion.

Proof Without Words: Sums of Triangular Numbers
Roger B. Nelsen
231
A visual proof that the sum of the first n triangular numbers is n(n+1)(n+2)/6.

On the Metamorphosis of Vandermonde's Identity
Don Rawlings and Lawrence Sze
232-238
In recent times, monomial pollination has become highly controversial. Critics are quick to point out that, while raising expectations, the procedure has a dismal failure rate. Undeterred by the critics, we pollinate the classical Vandermonde identity of combinatorics with a monomial analogous to the one appearing in the sum of the binomial theorem. The ensuing metamorphosis results in a truly remarkable matrix identity. Besides implying a host of amusing identities for binomial coefficients ranging from the well-known to the exotic, our matrix identity finds application in the area of system reliability.