American Mathematical Monthly -April 2000

April 2000

Knot Factoring
by Michael C. Sullivan
msulliva@math.sui.edu
A knot is just a closed loop in a three-dimensional space. Strangely enough, there is a notion of factoring knots and even a prime factoring theorem, due to Horst Schubert. We hope that our presentation of this classic result will be accessible to advanced undergraduates.

Evaluating Integrals Using Self-Similarity
by Robert S. Strichartz
str@math.cornell.edu
This paper gives a new elementary method for evaluating integrals of polynomials using only algebra and some basic properties of integrals, including self-similarity under dilation. The method has been used for some time to evaluate integrals on fractals, where it is the only method available. This paper shows how the method works in the context of ordinary calculus integrals, in a form that can be adapted easily for classroom use. It also illustrates the method for more exotic integrals on an interval with respect to Bernoulli measures, and discusses briefly the extension to self-similar fractals.

How Helical Can a Closed Twisted Space Curve Be?
by Joel L. Weiner
joel@math.hawaii.edu
A generalized helix is an unbounded curve and is characterized by the fact that the ratio of the curvature to the torsion is constant. How constant can that ratio be for a closed twisted space curve? A curve is twisted if its torsion is not identically zero and it necessarily does not lie in a plane. Examples show that there exist closed twisted curves for which the maximum and minimum values of the ratio are arbitrarily close. However, we show that the total variation of the ratio for such curves is bounded below by 4. Examples show that there exist closed twisted space curves for which the total variation of the ratio is greater than 8 but arbitrarily close to 8.

Norms, Isometries, and Isometry Groups
by Chi-Kwong Li
ckli@math.wm.edu
This paper gives a gentle introduction to the theory of norms and illustrates how group theory can be applied to questions in linear algebra. In particular, the importance of studying different norms on a vector space is discussed, and a group theory scheme for studying isometry groups and some related problems is presented.

Periodic Tilings as a Dissection Method
by F. Aguilo, M. A. Fiol, and M. L. Fiol
matfag@mat.upc.es, fiol@mat.upc.es, ml.fiol@cc.uab.es
We discuss a simple method for obtaining geometrical equidecompositions, based on superposing two or more congruent periodic tessellations. As applications of the method, we derive some dissections related to the Pythagorean Theorem, and prove the Bolyai-Gerwin and Hadwiger-Glur theorems, concerning the equidecomposability of two polygonal regions with equal areas.

Notes

A Better Bound on the Variance
by Chandler Davis and Rajendra Bhatia
davis@math.toronto.edu, rbh@isid.isid.ac.in

An Elementary Proof on Location of Zeros
by J. N. Ridley
k036jnr@cosmos.wits.ac.za

Recounting the Rationals
by Herbert S. Wilf and Neil Calkin
wilf@math.upenn.edu, calkin@math.clemson.edu

A Note on Positively Spanning Sets
by Stephen E. Wright
wrightse@muohio.edu

Problems and Solutions

Reviews

Basic Calculus: From Archimedes to Newton to its Role in Science
By Alexander J. Hahn

Reviewed by Julian F. Fleron
j_fleron@foma.wsc.mass.edu

Algebraic Aspects of Cryptography
By Neal Koblitz

Reviewed by Thomas W. Cusick
cusick@acsu.buffalo.edu

Editor's Endnotes