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Vol. 39, No. 3, pp. 174-256

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ARTICLES

**A Break for Mathematics: An Interview with Joe Gallian**

Deanna Haunsperger

174-190

Joe Gallian, President of the MAA, didn't grow up planning for a career in mathematics. A circuitous route through school, and an exceptionally good break (or lack thereof) for mathematics in a glass factory, started Joe on his way to a PhD. Through a series of serendipitous moments and his unquavering "just say yes" belief, Joe has become known widely for his esteemed work with his REU, his tireless energy with Project NExT, his longtime love of the Beatles, and his effervescent storytelling.

**An Alternate Approach to Alternating Sums: A Method to DIE For**

Arthur T. Benjamin and Jennifer J. Quinn

191-202

Positive sums count. Alternating sums match. Alternating sums of binomial coefficients, Fibonacci numbers, and other combinatorial quantities are analyzed using sign-reversing involutions. In particular, we Describe the quantity being considered, match positive and negative terms through an Involution, and count the Exceptions to the matching rule (the method of D.I.E.). Careful use of this technique often results in nice generalizations. Any sum arising from the Principle of Inclusion-Exclusion (P.I.E.), such as the number of derangements, can be understood using D.I.E. too.

**Dinner Tables and Concentric Circles: A Harmony of Mathematics, Music, and Physics**

Jack Douthett and Richard J. Krantz

203-211

How should men and women be seated around a dinner table to maximize conversation between members of the opposite sex? What can be said about the distribution of points around two concentric circles? How are the white and black keys on the piano keyboard organized? What spin configuration in the Ising model minimizes energy? These four problems have remarkably similar solutions.

**Squaring a Circular Segment**

Russell Gordon

212-220

Consider a circular segment (the smaller portion of a circle cut off by one of its chords) with chord length *c* and height *h* (the greatest distance from a point on the arc of the circle to the chord). Is there a simple formula involving *c* and *h* that can be used to closely approximate the area of this circular segment? Ancient Chinese and Egyptian records indicate the use of a formula based on a trapezoid to approximate this area, namely *h*(*c*+*h*)/2. Several centuries later, Archimedes discovered a formula (based on a triangle) that gives the exact area of a parabolic segment. Since a parabola can be used to approximate a circular arc, Archimedes' result yields 2*ch*/3 as another formula to approximate the area of the circular segment. A search for a better estimate, one that continues to rely on a quadratic function of *c* and *h*, reveals a much better approximation for this area than either of the ones mentioned thus farand generates some interesting elementary mathematics.

**Dependent Probability Spaces**

William F. Edwards, Ray C. Shiflett, and Harris Shultz

221-226

The mathematical model used to describe independence between two events in probability has a non-intuitive consequence called dependent spaces. The paper begins with a very brief history of the development of probability, then defines dependent spaces, and reviews what is known about finite spaces with uniform probability. The study of finite dependent spaces with non-uniform probabilities is then introduced. The paper concludes with an investigation of infinite spaces.

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CAPSULES

**From Mixed Angles to Infinitesimals**

Jacques Bair and Valérie Henry

230-233

In different ways, both Euclid (with his "horn angles") and Newton (with his "angles of contact") had a more general concept of angle than we do today. Giving a measure to these angles, as well as being of interest in its own right, leads to infinitesimal numbers.

**The Perimeter of a Polyomino and the Surface Area of a Polycube**

Wiley Williams and Charles Thompson

233-237

In this note, we develop a formula for the perimeter of a polyomino in terms of the number of tiles and interior vertices, and also a formula for the surface area of a polycube in terms of the number of cubes, interior edges, and interior vertices..

**The Cross Product as a Polar Decomposition**

Götz Trenkler

237-239

The cross product of two vectors a and b in 3-space can be given as a product **T**_{a}b, where **T**_{a} is a matrix that depends only on a. In this note, we show that this result is a natural consequence of a "polar decomposition" of the matrix **T**_{a}.

**The Right Theta**

William Freed and Athanasios Tavouktsoglou

The formula *θ* = arctan (*y/x*) gives the angle associated with a point (*x,y*) in the plane, valid for |*θ*| θ|