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American Mathematical Monthly Contents—March 2013

Hang on to your hats for the first special issue of the Monthly in over 20 years! The issue celebrates the International Mathematics Summer School for Students that took place in July of 2011 at Jacobs University in Bremen, Germany. The issue is guest edited by Dierk Schleicher (Jacobs University) and Serge Tabachnikov (Pennsylvania State University) and includes papers from, among others, John Conway, Dmitry Fuchs, Étienne Ghys, Don Zagier, and Tadashi Tokieda. Readers will not want to miss the amazing hand-drawn illustrations in Tokieda's paper "Roll Models." We hope you enjoy perusing this amazing collection of papers; the Problem Section will be back in April. —Scott Chapman

Vol. 120, No. 3, pp.191-294.

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About the cover: The special issue is devoted to the International Mathematical Summer School for Students at Jacobs University that took place in Bremen in 2011. The Free Hanseatic city Bremen is in northwestern Germany. The cover depicts the market square with the Town Hall and St. Peter's Cathedral. The four main characters are the Bremen Town Musicians of the Grimm Brothers' fairy tale: the donkey, the dog, the cat, and the rooster. There are several mathematical objects on the cover, related to the articles inside the issue: the tail of the rooster can be found in D. Fuchs's article; the front and rear bicycle tracks (along with the bicycle itself) in R. Foote, M. Levi, and S. Tabachnikov's article; and the sheet music in J. Conway's article. On the center of Bremen's market square, the Roland statue has bravely been overlooking the town center for more than six centuries—until March 2013 when the Bremen Town Musicians returned and played their music again.


On Unsettleable Arithmetical Problems

John H. Conway

It has long been known that there are arithmetic statements that are true but not provable, but it is usually thought that they must necessarily be complicated. In this paper, I shall argue that these wild beasts may be just around the corner.

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Tractrices, Bicycle Tire Tracks, Hatchet Planimeters, and a 100-year-old Conjecture

Robert Foote, Mark Levi, and Serge Tabachnikov

We study a simple model of bicycle motion: A bicycle is a segment of fixed length that can move in the plane so that the velocity of the rear end is always aligned with the segment. The same model describes the hatchet planimeter, a mechanical device for approximately measuring the area of plane domains.

The trajectory of the front wheel and the initial position of the bicycle uniquely determine its motion and its terminal position; the monodromy map sending the initial position to the terminal one arises. This circle mapping is a Möbius transformation, a remarkable fact that has various geometrical and dynamical consequences. Möbius transformations belong to one of three types: elliptic, parabolic, and hyperbolic. We describe a proof of a 100-year-old conjecture: If the front wheel track of a unit bike is an oval with area at least pi, then the respective monodromy is hyperbolic.

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Evolutes and Involutes of Spatial Curves

Dmitry Fuchs

We study evolutes and involutes of space curves.

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Intersecting Curves (Variation on an Observation of Maxim Kontsevich)

Étienne Ghys

Consider the graphs of n distinct polynomials of a real variable intersecting at some point. In the neighborhood of this point, the qualitative picture is described by some permutation of $$\{1,\dots, n\}$$. We describe the permutations that occur in such a situation.

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New Looks at Old Number Theory

Aimeric Malter, Dierk Schleicher, and Don Zagier

We present three results of number theory that all have classical roots, but also modern aspects. We show how to (1) systematically count the rational numbers by iterating a simple function, (2) find a representation of any prime congruent to 1 modulo 4 as a sum of two squares by using simple properties of involutions and pairs of involutions, and (3) find counterexamples to Euler's conjecture that a fourth power can never be the sum of three fourth powers by using properties of quadratic polynomials with rational coefficients.

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Roll Models

Tadashi Tokieda

These notes attempt a case study of applied mathematics for beginning students via problems of rolling. They do so by pointing out diverse surprising phenomena, many of which the reader can try at home, modeling them, and testing the limits of these models. One message is that rolling, because it tightly coordinates different modes of motion, tends to be more exactly solvable than meets the eye. Another message is that the thrill of applied mathematics is not in how difficult the mathematics is, but rather in what diversity of surprises in one's own experience one can discover, then understand.

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Mathematical Omnibus: Thirty Lectures on Classic Mathematics. By Dmitry Fuchs and Serge Tabachnikov. Reviewed by Rade T. Živaljević.

An Invitation to Mathematics: From Competitions to Research. Dierk Schleicher and Malte Lackmann, editors. Reviewed by Pieter Moree.

Dimensions. A mathematical video. By Jos Leys, Étienne Ghys, and Aurélien Alvarez. Reviewed by Christoph Pöppe.

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