# American Mathematical Monthly -January 2004

## January 2004

History or Heritage? An Important Distinction in Mathematics and for Mathematics Education
by Ivor Grattan-Guinness
i.grattan-guinness@mdx.ac.uk
Both mathematicians and historians are interested in the mathematics, but often in markedly different ways. The mathematician’s normal attention to history is with heritage; that is, how did we get here? Old results are modernized in order to show their current place; but the historical context is ignored and thereby often distorted. By contrast, the historian is concerned with what happened in the past, whatever be the modern situation. Each approach is perfectly legitimate, but they are often confused. The difference between them is discussed, with some emphasis given to consequences for mathematics education.

Positive Rational Solutions to xy = ymx : A Number-Theoretic Excursion
by Michael A. Bennett and Bruce Reznick
bennett@math.ubc.edu, reznick@math.uiuc.edu
The equation of the title has, over the years, been studied by Euler, Bernoulli, and countless others. In its simplest form, it provides a nice example of a Diophantine equation whose solution in rational numbers is an elementary, yet nontrivial exercise. In general, however, as we illustrate in this article, such equations lead quickly to much deeper water. We show how powerful tools from transcendental number theory shed light on these problems and reinforce, for the umpteenth time, the adage that there is often more to an equation than meets the eye.

Prince Rupert’s Rectangles
by Richard P. Jerrard and John E. Wetzel
jerrard@uiuc.edu, j-wetzel@uiuc.edu
In the late seventeenth century, Prince Rupert won a wager that a hole could be made in one of two equal cubes large enough for the other cube to slide through. A century later Nieuwland determined the largest cube that can pass through a cube of unit side by finding the largest square that fits in the cube. More generally, what is the largest rectangular box of given shape that can pass through a suitable hole in a unit cube? We answer this question by determining the largest rectangle of prescribed shape that fits in the cube.

Creating More Convergent Series
by Steven G. Krantz and Jeffery McNeal
sk@math.wustl.edu, mcneal@math.ohio-state.edu
We study permutations ? of the positive integers N with the properties:
(i) if Σj aj converges (conditionally), then Σ aσ(j) converges;
(ii) there exists a divergent series Σj bj such that Σj bσ(j) converges.
Such permutations turn out to be plentiful, and have many remarkable properties.

Variations on a Theme in Paper Folding
by Burkard Polster
When first encountered, the Hilton-Pedersen paper-folding algorithm for constructing rational angles and regular star polygons produces a magical "AHA!" effect similar to the one produced by a Möbius strip cut in half. In this article, we summarize this construction in a way that enables us to describe a number of related striking paper-folding constructions as alternative geometrical front-ends to the sound mathematical base created by Hilton and Pedersen.

Problems and Solutions

Notes

An Elementary Proof of the Quadratic Reciprocity Law
by Sey Y. Kim
sey.kim@kcl.ac.uk

Matrices Which Take a Given Vector into a Given VectorÂ—Revisited
by Götz Trenkler
trenkler@statistik.uni-dortmund.de

Finding ζ(2p) from a Product of Sines
by Thomas J. Osler
osler@rowan.edu

A Very Simple and Elementary Proof of a Theorem of Ingelstam
by S.H. Kulkarni
shk@iitm.ac.in

The Early History of the Ham Sandwich Theorem
by William A. Beyer
beyer@lanl.gov

Roots Appear in Quanta
by Alexander R. Perlis
aprl@math.arizona.edu

Reviews

The Other End of the Log: Memoirs of an Education Rebel
by Stephen S. Willoughby
Reviewed by Anthony Ralston
ar9@doc.ic.ac.uk

Imagining Numbers: (Particularly the Square Root of Minus Fifteen).
by Barry Mazur
Reviewed by Gerald B. Folland
folland@math.washington.edu

Abel’s Proof: An Essay on the Sources and Meaning of Mathematical Unsolvability.
by Peter Pesic
Reviewed by Gerald B. Folland
folland@math.washington.edu

The Riemann Hypothesis: The Greatest Unsolved Problem in Mathematics.
by Karl Sabbagh
Reviewed by Gerald B. Folland
folland@math.washington.edu

Telegraphic Reviews