In April’s Monthly, we aren’t foolin’! Learn about the dynamics of triangles in the plane in Richard Montgomery’s “The Three-Body Problem and the Shape Sphere.” In “Cyclic Evasion in the Three Bug Problem,” Maxim Arnold and Vadim Zharnitsky offer us a new and short proof of the three bug problem. Will Traves reviews Jürgen Richter-Gebert’s “Perspectives in Projective Geometry: A Guided Tour Through Real and Complex Geometry,” and our Problem Section will be waiting once you are finished grading Final Exams. Stay tuned for the May issue when Donald Saari present us with the thought provoking “Mathematics and the “Dark Matter Puzzle." - *Scott T. Chapman, Editor*

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Volume 122, Issue 04, pp. 297 - 402

## Table of Contents

## Articles

### The Three-Body Problem and the Shape Sphere

Richard Montgomery

The three-body problem defines dynamics on the space of triangles in the plane. The shape sphere is the moduli space of oriented similarity classes of planar triangles and lies inside shape space, a Euclidean three-space parameterizing oriented congruence classes of triangles. We derive and investigate the geometry and dynamics induced on these spaces by the three-body problem. We present two theorems concerning the three-body problem whose discovery was made through the shape space perspective.

To purchase the article from JSTOR: http://dx.doi.org/10.4169/amer.math.monthly.122.04.299

### A Fine Rediscovery

Rebekah Ann Gilbert

This article explores the history of the two results in integer partitions known as Stanley’s theorem and Elder’s theorem. While history has credited Richard Stanley with the discovery of the results, we note that Nathan Fine had established these results among a host of other partition identities over a decade earlier. In tribute to Fine, analogues in the sets of odd partitions and distinct partitions are presented.

To purchase the article from JSTOR: http://dx.doi.org/10.4169/amer.math.monthly.122.04.322

### Curvature for Polygons

Juliá Cufí, Agustí Reventós, and Carlos J. Rodríguez

Using a notion of curvature at the vertices of a polygon, we prove an inequality involving the length of the sides of the polygon and the radii of curvature at the vertices. As a consequence, we obtain a discrete version of Ros’ inequality.

To purchase the article from JSTOR: http://dx.doi.org/10.4169/amer.math.monthly.122.04.332

### The Polynomials of Mahler and Roots of Unity

Karl Dilcher and Larry Ericksen

Kurt Mahler in 1982 studied a special sequence of very sparse (0,1)-polynomials which have only powers of 2 as exponents. In this paper we study divisibility properties of these polynomials by certain cyclotomic polynomials and prove an explicit version of a result that was given only implicitly by Mahler. We also consider the distribution of real and complex noncyclotomic zeros, improving some of Mahler’s results. Then we show that the derivatives of the polynomials of Mahler have all their zeros inside the unit circle. We conclude this paper with some further remarks and open questions.

To purchase the article from JSTOR: http://dx.doi.org/10.4169/amer.math.monthly.122.04.338

### A New Metric on Spherical f-Tilings

Altino F. Santos

The continuous deformation of any spherical isometric folding into the standard spherical folding *f*_{s}, defined by *f*_{s}*(x, y, z) = (x, y, |z|)*, has been an open problem since 1989. Some relations between the deformation of spherical isometric foldings and the deformation of spherical f-tilings are analyzed, as they are closely related. The natural way to deform f-tilings is based on the Hausdorff metric on compact sets. However, this metric does not induce a continuous deformation on its associated isometric foldings. A new metric on spherical f-tilings will be introduced and a new contribution to the deformation of isometric foldings (via deformation of f-tilings) will be given.

To purchase the article from JSTOR: http://dx.doi.org/10.4169/amer.math.monthly.122.04.354

### Absolute Continuity of a Function and Uniform Integrability of its Divided Differences

Patrick M. Fitzpatrick and Brian R. Hunt

We describe a proof of the fundamental theorem of calculus for the Lebesgue integral, based on the following result: A real-valued function *f* on a compact interval is absolutely continuous if and only if its family of divided difference functions, {*x* ↦ [*f*(*x* + *h*) − *f*(*x*)]/*h*}_{0<h≤1}, is uniformly integrable. The Vitali convergence theorem states that, for a sequence of uniformly integrable functions, pointwise convergence implies integrability of the limit function and convergence of the integrals. Our characterization of absolute continuity, together with Vitali's theorem, shows that the fundamental theorem for absolutely continuous functions follows simply by passing to the limit in its discrete formulation.

To purchase the article from JSTOR: http://dx.doi.org/10.4169/amer.math.monthly.122.04.362

## Notes

### Face Numbers of Down-Sets

Michał Adamaszek

We compare various viewpoints on down-sets (simplicial complexes), illustrating how the combinatorial inclusion-exclusion principle may serve as an alternative to more advanced methods of studying their face numbers.

To purchase the article from JSTOR: http://dx.doi.org/10.4169/amer.math.monthly.122.04.367

### Generalizing Wallis’ Formula

Dirk Huylebrouck

The present note generalizes Wallis’ formula, , using the Euler–Mascheroni constant *γ* and the Glaisher–Kinkelin constant *A*.

To purchase the article from JSTOR: http://dx.doi.org/10.4169/amer.math.monthly.122.04.371

### A New Geometric Proof for Morley’s Theorem

Mehmet Kilic

This article gives a new and purely geometric proof for Morley’s theorem.

To purchase the article from JSTOR: http://dx.doi.org/10.4169/amer.math.monthly.122.04.373

### Cyclic Evasion in the Three Bug Problem

Maxim Arnold and Vadim Zharnitsky

In this note, we present a simple proof that three bugs involved in cyclic evasion converge to an equilateral triangle configuration. The approach relies on an energy-type estimate that makes use of a new inequality for the triangle.

To purchase the article from JSTOR: http://dx.doi.org/10.4169/amer.math.monthly.122.04.377

### A Short Proof of the Bradley Theorem

Ðorđe Baralic'

We give an elegant proof of the Bradley theorem. The theorem is an extension of the classical Carnot’s theorem for a conic intersecting the sides of a triangle. Although formulated in clear Euclidean language, Bradley’s theorem is a purely projective result.

To purchase the article from JSTOR: http://dx.doi.org/10.4169/amer.math.monthly.122.04.381

### Linear Recurrences via Probability

Byron Schmuland

The long run behavior of a linear recurrence is investigated using standard results from probability theory.

To purchase the article from JSTOR: http://dx.doi.org/10.4169/amer.math.monthly.122.04.386

## Problems and Solutions

To purchase the article from JSTOR: http://dx.doi.org/10.4169/amer.math.monthly.122.04.390

## Book Review

*Perspectives in Projective Geometry. A Guided Tour Through Real and Complex Geometry *By Jürgen Richter-Gebert

Reviewed by Will Traves

To purchase the article from JSTOR: http://dx.doi.org/10.4169/amer.math.monthly.122.04.398

## MathBits

### A Base Conversion Surprise

Vadim Ponomarenko

### A New Infinite Series Representation of Ink

Hidefumi Katsuura

### Solution to A Base Conversion Surprise

Vadim Ponomarenko

### 100 Years Ago This Month in the American Mathematical Monthly

Edited by Vadim Ponomarenko