May flowers bloom in the Monthly! Our lead article delves into the interesting aspects of the mathematics of "dark matter." Don't miss Donald Saari's explanation of these issues in his paper "Mathematics and the "Dark Matter" Puzzle." What is the probability that a subtree picked at random from a complete graph is a spanning tree? Find the surprising answer in Gary Gordon (and his 3 co-author's) paper "Pick a Tree - any Tree." Gerry Folland reviews "*A History in Sum: 150 Years of Mathematics at Harvard **(1825-1975)" *by Steve Nadis and Shing-Tung Yau. Have some spare time on your hands now that classes are finished? Try our Problem Section. Stay tuned for June/July when Jason Siefken and Dennis Epple show us how to play "Collapse." - *Scott T. Chapman, Editor*

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Volume 122, Issue 05, pp. 403 - 514

## Table of Contents

## Articles

### Mathematics and the “Dark Matter” Puzzle

Donald G. Saari

Surprisingly, aspects of the compelling mystery of “dark matter”—the mysterious, undiscovered material that supposedly consumes most of a spiral galaxy’s mass—are mathematical issues rather than astronomical ones. When analyzed from the perspective of mathematics, doubt is cast on standard predictions about the existence of huge amounts of this material.

To purchase the article from JSTOR: http://www.jstor.org/stable/10.4169/amer.math.monthly.122.5.407

### Pick a Tree – Any Tree

Alex Chin, Gary Gordon, Kellie MacPhee, and Charles Vincent

A subtree is picked at random from the collection of all subtrees of a complete graph. What is the probability the subtree spans? We find the surprising answer this question and a few closely related questions.

To purchase the article from JSTOR: http://www.jstor.org/stable/10.4169/amer.math.monthly.122.5.424

### Generalizations of Wilson’s Theorem for Double-, Hyper-, Sub- and Superfactorials

Christian Aebi and Grant Cairns

We present generalizations of Wilson’s theorem for double factorials, hyperfactorials, subfactorials, and superfactorials.

To purchase the article from JSTOR: http://www.jstor.org/stable/10.4169/amer.math.monthly.122.5.433

### A Simple Computation of ζ (2k)

Óscar Ciaurri, Luis M. Navas, Francisco J. Ruiz, and Juan L. Varona

We present a new simple proof of Euler’s formulas for ζ(2k), where k = 1, 2, 3,…. The computation is done using only the defining properties of the Bernoulli polynomials and summing a telescoping series. The same method also yields integral formulas for ζ(2k + 1).

To purchase the article from JSTOR: http://www.jstor.org/stable/10.4169/amer.math.monthly.122.5.444

### Using Rolling Circles to Generate Caustic Envelopes Resulting from Reflected Light

Jeffrey A. Boyle

Given any smooth plane curve α (s) representing a mirror that reflects light the usual way and any radiant light source at a point in the plane, the reflected light will produce a caustic envelope. For such an envelope we show that there is an associated curve β (s) and a family of circles Cs that roll on β (s) without slipping such that there is a point on each circle that will trace the caustic envelope as the circles roll. For a given curve α (s) and for all radiants at infinity there is a single curve β (s) and family of circles Cs that roll on β (s) so that the different points on Cs will simultaneously trace out, as the circles roll, all caustic envelopes from these radiants at infinity. We explore many classical examples using this method.

To purchase the article from JSTOR: http://www.jstor.org/stable/10.4169/amer.math.monthly.122.5.452

### A Porism for Cyclic Quadrilaterals, Butterfl y Theorems,and Hyperbolic Geometry

Ivan Izmestiev

If there exists a cyclic quadrilateral whose sides go through the given four collinear points, then there are infinitely many such quadrilaterals inscribed in the same circle. We give two proofs of this porism: one based on cross-ratios, the other on compositions of hyperbolic isometries.

To purchase the article from JSTOR: http://www.jstor.org/stable/10.4169/amer.math.monthly.122.5.467

## Notes

### Side Lengths of Equiangular Polygons (as seen by a coding theorist)

Maria Bras-Amorós and Marta Pujol

A tuple of positive real numbers is said to be *equiangular* if there is an equiangular polygon with consecutive side lengths . It is well known that *a* is equiangular if and only if the polynomial vanishes at . Here we dispense with complex numbers and borrow an idea from the theory of cyclic codes to prove that *a* is equiangular if and only if is divisible by .

To purchase the article from JSTOR: http://www.jstor.org/stable/10.4169/amer.math.monthly.122.5.476

### Relationships Between the First Four Moments

Iosif Pinelis

Best possible bounds on the third moment of a random variable are given in terms of the first, second, and fourth moments.

To purchase the article from JSTOR: http://www.jstor.org/stable/10.4169/amer.math.monthly.122.5.479

### Eigenvalues of Real Symmetric Matrices

Meinolf Geck

We present a proof of the existence of real eigenvalues of real symmetric matrices which does not rely on any limit or compactness arguments, but only uses the notions of “sup”, “inf”.

To purchase the article from JSTOR: http://www.jstor.org/stable/10.4169/amer.math.monthly.122.5.482

### A One-Formula Proof of the Nonvanishing of *L*-Functions of Real Characters at 1

Bogdan Veklych

We present a simple analytic proof that L-functions of real nonprincipal Dirichlet characters are nonzero at 1.

To purchase the article from JSTOR: http://www.jstor.org/stable/10.4169/amer.math.monthly.122.5.484

### The Volume of the Convex Hull of a Body and its Homothetic Copies

Jesús Jerónimo Castro

In this note, we prove the following result. Let K ⊂ ℝd be a convex body with the origin O in its interior. If there is a number λ ∈ (0, 1) such that the n-dimensional volume of the convex hull of the union of K with the translates of λ K, by a vector x, depends only on the Euclidean norm of x, then K is a Euclidean ball.

To purchase the article from JSTOR: http://www.jstor.org/stable/10.4169/amer.math.monthly.122.5.486

### Explaining a Mysterious Maximal Inequality — and a Path to the Law of Large Numbers

J. Michael Steele

In 1964 A. Garsia gave a stunningly brief proof of a useful maximal inequality of E. Hopf. The proof has become a textbook standard, but the inequality and its proof are widely regarded as mysterious. Here we suggest a straightforward first step analysis that may dispel some of the mystery. The development requires little more than the notion of a random variable, and, the inequality may be introduced as early as one likes in a graduate probability course. The benefit is that one gains access to a proof of the strong law of large numbers that is pleasantly free of technicalities or tricky ideas.

To purchase the article from JSTOR: http://www.jstor.org/stable/10.4169/amer.math.monthly.122.5.490

### The Brocard Angle and a Geometrical Gem from Dmitriev and Dynkin

Ádám Besenyei

In a celebrated paper on the eigenvalues of special matrices, N. Dmitriev and E. Dynkin formulated and proved a nice geometrical lemma. This lesser-known gem provides a simple proof of the maximal value of the Brocard angle and yields also the solution to some challenging problems from the past few decades.

To purchase the article from JSTOR: http://www.jstor.org/stable/10.4169/amer.math.monthly.122.5.495

## Problems and Solutions

To purchase the article from JSTOR: http://www.jstor.org/stable/10.4169/amer.math.monthly.122.5.500

## Book Review

*A History in Sum: 150 Years of Mathematics at Harvard(1825-1975)*By Steve Nadis and Shing-Tung Yau

Reviewed by Gerald B. Folland

To purchase the article from JSTOR: http://www.jstor.org/stable/10.4169/amer.math.monthly.122.5.508

## MathBits

### A New Proof of the Pythagorean Theorem

Nam Gu Heo

### A One-Line Proof of the Infinitude of Primes

Sam Northshield

### An Upper Bound for Some Coloring Problem

Janusz Januszewski

### Using Infinitesimals to Differentiate Secant and Tangent

Michael Hardy

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

Edited by Vadim Ponomarenko