An Elementary Introduction to the Hopf Fibration
David W. Lyon
This article about the Hopf fibration began with a graduate textbook exercise to prove that fibers of the Hopf map are linked. Struck by the beauty of arrangements of fiber circles, the author came upon a way to describe the Hopf fibration using mathematics accessible to undergraduate students. The article introduces the quaternions to describe and study algebraic aspects of the Hopf fibration and rotations of 3-dimensional space. The article also examines geometry of Hopf fibers using stereographic projection. A set of structured exercises called Investigations suggests many possibilities for student independent study.
C. S. Peirce and the Bell Numbers
Even C. S. Peirce's admirers admit that wading through his prose can be an unpleasant and unprofitable experience. Nonetheless, this essay encourages mathematically inclined readers to sample Peirce. It does so by dangling a characteristic gem: a contribution to a fundamental area of combinatorics that Peirce offers as an aside in a famous, but famously impenetrable, paper on logic.
The Dinner-Diner Matching Problem
Barbara Haas Margolius
This note addresses a matching problem or problem with restricted position. Such problems have a long history in classical probability and are often described as card-matching problems. At the spring 2000 meeting of the Ohio section of the MAA in Huntington, West Virginia, there was a banquet for NExT fellows, untenured faculty in the section's faculty development program. Each attendee had selected a dinner from among four entrees. The following orders had been placed by the twenty-two guests: 1 pasta vegetarian, 8 chicken cordon bleu, 6 prime rib, and 7 filet of sole dinners. None of the guests could say with certainty what they had ordered for dinner, leaving the server greatly distressed.
Dinner-Diner Matching Problem:
If the dinners are served randomly, what is the probability distribution of the number of diners who are served what they ordered?
A Natural Generalization of the Win-Loss Rating System
How does one factor a team's schedule strength into its final standing? We offer a mathematically simple and natural way to do this that highlights some of the ideas covered in an elementary linear algebra course. Students learning about matrix multiplication, eigenvectors and eigenvalues, or even limits in a calculus course will find an interesting application to supplement their study.
Volume, Surface Area, and Harmonic Mean
Paul Fjelstad and Ivan Ginchev
The volume-area-derivative relationship dV/dr = A for a ball of radius r is a rather striking one when one first comes across it. Accordingly, it is rather natural to consider generalizations, as was done in a Note in this Magazine, 70, (1997), pp. 365-371, first for n-dimensional polyhedra that circumscribe n-dimensional spheres. When the circumscription fails, the authors of the Note suggest saving the relationship by pushing out the polyhedra's faces a distance of e, while noting that this does not produce a family of similar objects. Here an alternative parameter h is introduced, in which the harmonic mean plays a role. The consequence is a family of similar objects and the rescue of the volume-area-derivative relationship, namely dV(h)/dh = A(h), where h = nV/A. This definition for h opens up the possibility for a sweeping generalization, for it applies not only to families of similar n-dimensional polyhedra, but to any family of similar objects that have some n-dimensional content V and some (n-1)-dimensional content A.
Markov Chains for the RISK Board Game Revisited
Jason A. Osborne
In a wonderful article in the Magazine about the board game RISK, Tan (1997) presented a mathematical framework for calculation of probabilities involved in a battle for a given country. The calculations for the Markov chain presented in the article are off a bit due to misspecification of the transition probability matrices. This note provides a correction and a few extended computations that result in revised strategic recommendations for the well-informed player.
PROOFS WITHOUT WORDS
The Cube as an Arithmetic Sum
The cube of a number is exhibited as an arithmetic progression that begins with a triangular number.
Visualizing a Nonmeasurable Set
Saleem Watson and Arthur Wayman
We construct a nonmeasurable subset of the torus, similar to the well-known Vitali subset of the unit interval. Our set is "visual" in the sense that we can draw the elements used in constructing the set. The main ingredient in the construction is a continuous group homomorphism from R into the torus; its image can be visualized as a "coil" wound about the torus.
Nondifferentiability of the Ruler Function
The ruler function on the unit interval is examined in an introductory analysis course because of its counterintuitive properties: although continuous at each irrational and discontinuous at each rational, it nonetheless is Riemann integrable. Here we address its differentiability, proving that the ruler function is nowhere differentiable by means of an elementary proof that features a cameo appearance by none other than Euclid himself.
Convergence of p-series
Sydney H. Kung
One of the effective ways of proving the convergence of p-series is by rearranging the terms and comparing them with another series. A special grouping method of the terms of p-series shown here leads to an easy comparison with the geometric series for the case where p>1. Readers are encouraged to investigate the divergence of the series by using the same approach in the case where p<1. (Editor's Note: This is essentially the Cauchy Condensation Test.)
A Classification of Matrices of Finite Order over C, R, and Q
Can one find a 2x2 rational matrix of any order? We show that a rational matrix of finite order is necessarily similar to a direct sum of companion matrices of cyclotomic polynomials. As a consequence we see that a 2x2 rational matrix can have orders only 1, 2, 3, 4, or 6. We also show that a real matrix of finite order is necessarily similar to a direct sum of 2x2 rotation and reflection matrices. A complex matrix of finite order is necessarily similar to a diagonal matrix whose diagonal entries are complex roots of unity. These classification theorems are obtained from a blend of several topics of undergraduate algebra and number theory.
The Extra Distance in an Outer Lane of a Running Track
Elliott A. Weinstein
This answers the oft-asked question of how much extra distance over that of the innermost lane is added per lap to one's workout when using an outer lane of a running track.