May 2011 Contents
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In the May issue of The College Mathematics Journal, Bonnie Gold explains how your philosophy of mathematics impacts your teaching and Andrew Simoson estimates in various ways, historical and otherwise, the magnitude of the earth's bulge at the equator. In addition, among other items, there are articles on art gallery theorems, counting subgroups in a direct product of cyclic groups, and change ringing. A Classroom Capsule describes a form of induction, called continuity induction, suitable for proving fundamental theorems in analysis. And Martin Ericson contributes the names of some legendary mathematical movies (for example the notorious ‘Texas Chain Rule Massacre').
How Your Philosophy of Mathematics Impacts Your Teaching
Even those with no interest in the philosophy of mathematics take positions on a range of philosophical issues when they teach mathematics. Some of these issues are intimately related to common student confusions. These include the nature of mathematical objects and relations, such as the real numbers and equality; how we come to conclude a mathematical statement is true; how mathematical objects relate to those in the physical world; and how people learn mathematics.
Newton’s Radii, Maupertuis’ Arclengths, and Voltaire’s Giant
Andrew J. Simoson
Given two arclength measurements along the perimeter of an ellipse—one taken near the long diameter, the other taken anywhere else—how do you find the lengths of major and minor axes? This was a problem of great interest from the time of Newton’s Principia until the mid-eighteenth century when France launched twin geodesic expeditions—one to the equator near Quito, the other to the Finnish arctic led by Maupertuis—so as to determine whether the earth was lemon-shaped, as the French Academy long contended, or like a mandarin orange as Newton promised. We give a simplified version of Newton’s argument, and show how an elliptical profile model for the earth’s shape together with an arclength measurement determines the amount of flattening of the earth at the poles. We conclude by speculating why Voltaire made his giant Micromégas exactly 23 miles tall.
Guards, Galleries, Fortresses, and the Octoplex
T. S. Michael
The art gallery problem asks for the maximum number of stationary guards required to protect the interior of a polygonal art gallery with n walls. This article explores solutions to this problem and several of its variants. In addition, some unsolved problems involving the guarding of geometric objects are presented.
The random breakage of a rod into unit lengths
Joe Gani and Randall Swift
In this article we consider the random breakage of a rod into L elements of unit length and present a Markov chain based method that tracks intermediate breakage configurations. The probability of the time to final breakage for L = 3, 4, 5 is obtained and the method is shown to extend in principle, beyond L = 5.
An Arithmetic Metric
This work introduces a distance between natural numbers not based on their position on the real line but on their arithmetic properties. We prove some metric properties of this distance and consider a possible extension.
Counting Subgroups in a Direct Product of Finite Cyclic Groups
We calculate the number of subgroups in a direct product of finite cyclic groups by applying the fundamental theorem of finite abelian groups and a well-known structure theorem due to Goursat. We also suggest ways in which the results can be generalized to a direct product of arbitrary finite groups.
An Application of Group Theory to Change Ringing
Michele Intermont and Aileen Murphy
After providing some background on change ringing, this paper introduces a question that arises in the systematic ringing of church bells and uses group theory to solve it.
Teaching Tip: Practice Integration on Problem Triplets
Triples of integrals with nearly identical integrands but requiring very different integration methods are suggested for student integration practice.
Teaching Tip: A Slippery Slope
Raymond A. Beauregard
A calculus problem demonstrates a potential confusion with units.
Using Continuity Induction
Here is a technique for proving the fundamental theorems of analysis that provides a unified way to pass from local properties to global properties on the real line, just as ordinary induction passes from local implication (if true for k, the theorem is true for k + 1) to a global conclusion in the natural numbers.
PROBLEMS AND SOLUTIONS
Crossing the Equals Sign by Marion Deutche Cohen
Reviewed by Annalisa Crannell
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