This issue of the Mathematics Magazine highlights the range of diverse articles we publish. David Nash and Jonathan Needleman ask, and answer, the question: When are finite projective planes magic? Rob Poodiack looks at trigonometry and calculus based on hyperellipses instead of circles and touches on Danish design. Take a look at the symmetry Jeffrey Lawson and Matthew Rave use to analyze geometric phase in dynamical systems, relating the mathematics to Chopin, amusement park rides, and falling cats.
Further in the issue, Brian Conrey, James Gabbard, Katie Grant, Andrew Liu, and Kent Morrison query, "How rare are intransitive dice?" They make conjectures about the frequency of certain types of dice behavior and prove asymptotic results for the frequency of "one-step" dice to be intransitive.
—Michael A. Jones, Editor
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Vol. 89, No. 2, pp 80 – 155
When Are Finite Projective Planes Magic?
David A. Nash and Jonathan Needleman
We study a generalization of magic squares, where the entries come from the natural numbers, to magic finite projective planes, where the entries come from Abelian groups. For each finite projective plane we demonstrate a small group over which the plane can be labeled magically. In the prime order case we classify all groups over which the projective plane can be made magic.
To purchase from JSTOR: 10.4169/math.mag.89.2.83
Squigonometry, Hyperellipses, and Supereggs
Robert D. Poodiack
A superegg is the solid of revolution for a hyperellipse, an ellipse with squarish corners. The two objects became the basis of a design revolution of sorts in the 1960s, as practiced by the Danish mathematician and poet Piet Hein. We use an analog of trigonometry called squigonometry to produce a set of constants akin to the well-known and customary π and then find formulas using these constants for the area of a hyperellipse and the volume of Hein′s superegg.
To purchase from JSTOR: 10.4169/math.mag.89.2.92
Proof Without Words: Infinitely Many Almost-Isosceles Pythagorean Triples Exist
Roger B. Nelsen
Wordlessly, we show that there are infinitely many Pythagorean triples with consecutive integers as legs and infinitely many Pythagorean triples with consecutive triangular numbers as legs.
To purchase from JSTOR: 10.4169/math.mag.89.2.103
Spacewalks and Amusement Rides: Illustrations of Geometric Phase
Jeffrey Lawson and Matthew Rave
Geometric phase in a dynamical system can be visualized as the interplay between two periodic functions which go in and out of “synch.” Using illustrations of a boy′s walk in space and a dizzying fun park ride, we demonstrate that in certain simple mechanical systems we can compute geometric phase directly from a symmetry—we don′t even need to solve the system of differential equations. We can also use geometric phase to explain how cats (almost) always land on their feet. We conclude with an interpretation of geometric phase in terms of the geometric notions of connection and curvature.
To purchase from JSTOR: 10.4169/math.mag.89.2.105
Bounds for the Representations of Integers by Positive Quadratic Forms
Kenneth S. Williams
Recent ground-breaking work of Conway, Schneeberger, Bhargava, and Hanke shows that to determine whether a given positive quadratic form F with integer coefficients represents every positive integer (and so is universal), it is only necessary to check that F represents all the integers in an explicitly given finite set S of positive integers. The set contains either nine or twenty-nine integers depending on the parity of the coefficients of the cross-product terms in F and is otherwise independent of F. In this article we show that F represents a given positive integer n if and only if F(y1, … , yk) = n for some integers y1, … , yk satisfying where the positive rational numbers ci are explicitly given and depend only on F. Let m be the largest integer in S (in fact m = 15 or 290). Putting these results together we have F is universal if and only if
To purchase from JSTOR: 10.4169/math.mag.89.2.122
Proof Without Words: van Schooten’s Theorem
We provide a simple visual proof of van Schooten′s theorem: given an equilateral triangle ABC with circumcircle, if point P is chosen on minor arc BC, then PA = PB + PC.
To purchase from JSTOR: 10.4169/math.mag.89.2.132
Brian Conrey, James Gabbard, Katie Grant, Andrew Liu, and Kent E. Morrison
We consider n-sided dice whose face values lie between 1 and n and whose faces sum to n(n + 1)/2. For two dice A and B, define A ≻ B if it is more likely for A to show a higher face than B. Suppose k such dice A1, … , Ak are randomly selected. We conjecture that the probability of ties goes to 0 as n grows. We conjecture and provide some supporting evidence that—contrary to intuition—each of the assignments of ≻ or ≺ to each pair is equally likely asymptotically. For a specific example, suppose we randomly select k dice A1, … , Ak and observe that A1 ≻ A2 ≻ … ≻ Ak. Then our conjecture asserts that the outcomes Ak ≻ A1 and A1 ≻ Ak both have probability approaching 1/2 as n → ∞.
To purchase from JSTOR: 10.4169/math.mag.89.2.133
To purchase from JSTOR: 10.4169/math.mag.89.2.144
Problems and Solutions
To purchase from JSTOR: 10.4169/math.mag.89.2.147
Riemann Hypothesis; Hunger Games; modular forms in context; coincidences
To purchase from JSTOR: 10.4169/math.mag.89.2.155