# American Mathematical Monthly - December 2003 ## DECEMBER 2003

When Does Water Find the Shortest Path Downhill? The Geometry of Steepest Descent Curves
by Daniel Drucker and Stephen A. Williams
drucker@math.wayne.edu and Williams@math.wayne.edu
Water flowing down a hill can reasonably be assumed to follow steepest descent curves. Such curves usually are not geodesics, but we determine exactly when they are. Surprisingly, there are surfaces all of whose steepest descent curves are geodesics; we classify them and explain how to construct them. We also determine when steepest descent curves are asymptotic or principal and give an example of a surface all of whose steepest descent curves are asymptotic. The Role of Logic in Teaching Proof
by Susanna Epp
sepp@condor.depaul.edu
Even simple mathematical proofs and disproofs are more logically complex than most mathematicians realize. Research by mathematics educators and cognitive psychologists supports the claim that the logical reasoning abilities needed to discover them are not widely present in the population. Two hypotheses are proposed to explain some of the reasons why so many students have difficulty with proof and disproof: differences between mathematical language and the language of everyday discourse, and the kinds of shortcuts and simplifications that have been part of students’ previous mathematical instruction. The article describes research about whether instruction can help students develop formal reasoning skills and suggests that such instruction can be successful when done with appropriate parallel development of transfer skills. The final sections discuss at what point the principles of logic should be introduced and give a variety of suggestions about how to teach them. Negative Theorems in Approximation Theory
by Allan Pinkus
pinkus@tx.technion.ac.il
We consider some negative results in approximation theory, what they tell us, and why they are important. The results we present all center about one theme, namely, that approximation processes are necessarily of limited capability. As approximation theory is concerned with the ability to approximate functions and processes by simpler and more easily calculated objects, it is important to know these limitations. We discuss three main categories of results, with a few elementary examples from each. The first category consists of inverse theorems, the second set of results is connected with limitations of linear processes, and the third group is concerned with n-width type results. Eudoxus Meets Cayley
by Richard E. Chandler, Carl D. Meyer, and Nicholas J. Rose
chandler@math.ncsu.edu, meyer@ncsu.edu, njrose@math.ncsu.edu
This article presents an intriguing marriage of classical geometry and the modern theory of stochastic matrices for the purpose of analyzing the limiting properties of subtriangles and supertriangles. The (p,q)-subtriangle of triangle ABC is triangle A'B'C' in which A', B', and C' divide the sides AB, BC, and CA, respectively, in the ratio p:q. The (p,q)2-subtriangle of ABC is the (p,q)-subtriangle of A'B'C'. An ancient result is that the (1,1)-subtriangle of ABC is similar to ABC. After proving that the (1,2)2-subtriangle of ABC is also similar to ABC, we address the question of whether the (1,n)n-subtriangle of ABC is similar to ABC for all natural numbers n. The answer to this question along with the analysis of limiting behavior of various iterated subtriangle procedures is accomplished by applying the theory of stochastic matrices, matrix exponentials, and the gamma function. A Localized Probabilistic Approach to Circuit Analysis
by Frank Blume
fblume@jbu.edu
The paper demonstrates that the standard method of analyzing resistor networks via systems of linear equations is equivalent to a probabilistic approach that translates resistances into transitional probabilities in such a way that electric currents in the circuit can be calculated from the unique invariant probability vector of the corresponding stochastic matrix. Notes

A Topological Puzzle
by Inta Bertuccioni
inta@magma.unil.ch

The Ascoli-Arzelà Theorem via Tychonoff’s Theorem
by David C. Ullrich
ullrich@math.okstate.edu

Klee’s Trigonometry Problem
by P.L. Kannappan
plkannappan@pythagoras.uwaterloo.ca

Green’s Theorem and the Fundamental Theorem of Algebra
by Paul Loya
paul@math.binghamton.edu

Mean Curvature and Asymptotic Volume of Small Balls
by Dominique Hulin and Marc Troyanov
Dominique.hulin@math.u-psud.fr, marc.troyanov@epfl.ch

Timing Is Everything: The French Connection
C.W. Groetsch
groetsch@uc.edu

Reviews

Weighing the Odds: A Course in Probability and Statistics
by David Williams
Reviewed by Ed Waymire
waymire@math.orst.edu

Drawbridge Up: MathematicsÂ—A Cultural Anathema
by Hans Magnus Enzensberger
Reviewed by Marion D. Cohen
mathwoman199436@aol.com

Classical and Quantum Computation
by A. Yu. Kitaev, A. H. Shen, and M. N. Vyalyi
Reviewed by Daniel Gottesman
dgottesman@perimeterinstitute.ca

Telegraphic Reviews

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