Awards for exemplary writing in MAA journals and magazines were handed out in August at MAA MathFest in Portland, Oregon. The winning articles (PDF) are now available online, along with the award citations explaining why each article merited an award.
Paul R. Halmos–Lester R. Ford Awards (presented to authors of articles of expository excellence published in The American Mathematical Monthly)
Will Traves
From Pascal’s Theorem to d-Constructible Curves
The American Mathematical Monthly, Vol. 120, no. 10, December 2013, pp. 901-915
Citation: Beginning with the history of the word syzygy, the author of this paper turns to Pascal’s Theorem: If six distinct points A, B, C, a, b, c lie on a conic, then the lines Ab, Bc, and Ca meet the lines aB, bC, and cA in three new collinear points. Pascal’s Theorem leads to deep but very natural questions about d-constructible curves. A curve S of degree t is d-constructible if there exist k = d + t red lines and k blue lines so that: the red lines meet the blue lines in k2 distinct points and dk of these points lie on a curve C of degree d and the remaining tk points lie on the curve S.
Traves grounds the reader firmly in the history and motivation of the problems in this area of algebraic geometry and leads us to an understanding of d-constructible curves and of the dimensions in which d-construction is dense. In a lucid and comprehensive exposition of the ideas stemming from Pappus’s Theorem on line arrangements, to its generalization to 6 points on a conic by Pascal, to a converse of Pascal’s Theorem by Braikenridge and Maclaurin, to Möbius’s generalization of Pascal’s Theorem, the reader is introduced to work by Eisenbud, Green, and Harris on the Cayley-Bacharach Theorem.
Traves includes an introduction to projective geometry with exercises for the beginner, Bézout’s Theorem, and the Zariski topology. This is material that can be technical and off-putting to the uninitiated. However, Traves embeds each idea in historical contexts that highlight connections and expose the underlying structure without trivializing or glossing over difficulties.
Tadashi Tokieda
Roll Models
The American Mathematical Monthly, Vol. 120, no. 9, November 2013, pp. 265-282
Citation: Paul Halmos claimed that the heart of mathematics is problem solving. This article exemplifies Halmos’s thesis by presenting 19 problems—but not problems of the usual variety! “Which way will it roll? . . . Problem 1: Make a guess. Have you made a guess? Now try the experiment.”
The author invites the reader to participate with pens and pot lids and golf balls and spools of thread as lab materials. Along the way, the reader delights in the numerous asides, such as the claim that the number 7 = 2 + 5 rarely appears in mechanics except in the context of the dimensionless moment of inertia (2/5) of a solid ball. Memorable wordings like “did they telephone each other” or “non-Drosophila biology” or “the time the ball takes to adjust to the shock” or “unprobable” decorate the chatty informal text.
The article illustrates that applied mathematics begins with surprising phenomena that lead to surprisingly simple principles (e.g., conserved quantities), which lead to more investigations. This paper fully realizes the goals of the Monthly: to “inform, stimulate, challenge, enlighten, and even entertain” its readers.
Jacques Lévy Véhel and Franklin Mendivil
Christiane’s Hair
The American Mathematical Monthly, Vol. 120, no. 3, March 2013, pp. 771-786
Citation: Beginning with a striking visualization of stacked Cantor sets that reminded the first author of his wife Christiane’s braided hair, Lévy Véhel and Mendivil guide the reader through an exploration of the geometric and measure-theoretic properties of these fractal sets.
The authors describe the constructions from different perspectives, using iterated function systems as well as ternary (and more general) expansions. While the horizontal cross-sections of their constructions are Cantor sets generated from the interval [0,1] with continuously changing gap parameters, the vertical view consists of many smooth strands, thus producing an appearance of a woman’s hair. They prove that not only is each strand a smooth curve, it is in fact real analytic. While these strands emanate from the points of [0,1], their structure is more complicated—the strands can be indexed by infinite binary sequences. For instance, the point x = 1/2 has two binary representations, corresponding to two different strands that originate from x = 1/2.
The authors explore various properties of these intriguing sets, such as the area between the strands and the Hausdorff dimension of the original set and generalizations of it, in a clear and accessible manner. The article also has a wonderful mix of calculations that a calculus student can understand along with deeper analysis to satisfy an inquiring mathematical audience.
Susan H. Marshall and Alexander R. Perlis
Heronian Tetrahedra Are Lattice Tetrahedra
The American Mathematical Monthly, Vol. 120, no. 2, February 2013, pp. 140-149
Citation: Heronian triangles are those whose side lengths and area are integer-valued. These seemingly simple objects have long been studied and written about, including a result from this journal by P. Yiu that they can be placed in the plane so the vertices have integer coordinates.
In this paper by Marshall and Perlis, the authors bring Yiu’s result to three dimensions by considering Heronian tetrahedra, those tetrahedra with integer side lengths, face areas, and volume. The authors begin by reviewing an alternate proof to Yiu’s result by J. Fricke that represents the locations of the vertices of the triangle as complex numbers and uses the arithmetic of Gaussian integers to find a rotation that moves the vertices onto the integer lattice.
Their review is organized in a straightforward manner, which then provides for a clear transition to the tetrahedron. By considering the vertices of the tetrahedron as quaternions and using various arithmetic results on Lipschitz-integral quaternions, the authors show that the previous argument can be adapted to yield the same result on the Heronian tetrahedron: that there is a rotation that moves the vertices onto the integer lattice. Especially nice are the concrete examples provided of this method to an explicit Heronian triangle and Heronian tetrahedron, letting the reader solidify his or her understanding in a very enjoyable way.