The College Mathematics Journal - May 1998

Contents for May 1998

In Love with Geometry
Daniel Pedoe

Dan Pedoe is an author of several influential books on various aspects of geometry: the three-volume monograph Methods of Algebraic Geometry with W. V. D. Hodge, expository works such as Circles: A Mathematical View, Geometry and the Visual Arts, and Japanese Temple Geometry Problems, with H. Fukagawa, and several textbooks. In this lively biographical memoir he recalls his early years and incidents from his time at Cambridge and other universities in the U. K., a decade at outposts of the vanishing British empire at Khartoum University and the University of Singapore, and later years at Purdue and the University of Minnesota. Pedoe's remimiscences include vignettes of Hodge, Russell, Zariski, Lefschetz, Weil, and Dyson, among many characters and places.

 

Studying the Cantor Dust at the Edge of Feigenbaum Diagrams
Aaron Klebanoff and John Rickert

The Feigenbaum diagram (or bifurcation diagram) of a 1-parameter family of maps on the real line shows the asymptotic states for a typical orbit, for a range of values of the parameter. However, it often happens that above a certain parameter value almost all orbits diverge to infinity; only a Cantor set of points remains invariant under the map and this Cantor set cannot be shown realistically on a computer screen. Modifying a method used for picturing Julia sets of complex maps, the authors propose what they call a divergence diagram that indicates the fractal dimension of the Cantor sets for parameter values beyond the range shown in the Feigenbaum diagram. Examples are given for the family of tent maps, the logistic maps, and the quadratic maps associated with the Mandelbrot set.

 

A Fresh Approach to the Singular Value Decomposition
Colm Mulcahy and John Rossi

The authors show how the singular value decomposition (SVD) can be naturally introduced near the end of a typical elementary linear algebra course. For invertible matrices the SVD is equivalent to the polar decomposition, a decomposition that is easy to motivate and prove . By using a computer algebra system students can quickly calculate the SVD even for noninvertible matrices, and they can see how the SVD leads to the useful pseudo-inverse of a matrix.

 

An Application of Elementary Group Theory to Central Solitaire
Arie Bialostocki

Central solitaire is played on a board that has 33 holes arranged in a symmetric cross, initially with a peg in each hole but the central one. A move consists of a horizontal or vertical jump of a peg over an adjacent one to land in an empty hole; the jumped peg then being removed from the board. The object of the game is to remove as many pegs as possible. By tiling the game board with the elements of the Klein 4-group in a simple pattern, the author shows that the game can end with a single peg on the board only if the peg is in one of five holes. Then, a simple sequence of moves is given for completing the game with a single peg any of these five locations.

 

Undersampled Sine Waves
Jean-Claude Derderian and Enriqueta Rodrigues-Carrington

To plot the graph of a function over an interval many graphing calculators simply divide the interval into equal subintervals, evaluate the function at the dividing points, and then plot the resulting points on the graph, perhaps connecting adjacent points by straight line segments. Undersampling occurs when the partition of the interval is too coarse to capture sudden fluctuations in the function's values. The authors discuss two types of errors in the graphs of sine waves that may arise from undersampling, which they call frequency reduction and amplitude modulation, and they clarify the source of these errors. Theorems and examples are given that show what the graph of a sine wave of a given frequency will look like, when it is produced by using a partition of the interval [0, ] into a given number of equal subintervals.

 

Classroom Capsules

Min Deng and Mary T. Whalen, The Mathematics of Cootie
In the children's game of "Cootie" each player tries to be the first to build a complete "cootie bug." Each of the bug's component parts is obtained by rolling a die. The body must be obtained first, which requires rolling a 1; then the head requires a 2. Thereafter, the remaining parts can be collected in any order: two eyes (each requires a 3), one nose (a 4), two antennae (each a 5) and the six legs (one 6 apiece). The authors find the expected number of rolls of a die needed to complete the game, using standard counting methods for finding the probabilities of events associated with independent repeated trials. Although the resulting formula for the expected number of rolls needed to complete the game of Cootie is cumbersome, it can easily be evaluated on a computer.

Michael Scott McClendon, Minimal Pyramids
What are the dimensions of the pyramid of least volume that can be circumscribed about a given sphere, if the pyramid's base is a regular n-gon? This optimization problem is solved by standard methods of calculus and the answer turns out, rather surprisingly, to be independent of n: the height of the minimal pyramid is twice the diameter of the sphere. Hence a similar result holds for cones circumscribed about a sphere.

Mike O'Leary, Taylor Polynomials for Rational Functions
Given a rational function defined at the origin, and any natural number n, students can use long division to write the given function as a polynomial of degree n plus a rational remainder. It is easily shown that this polynomial is the nth Taylor polynomial of the given rational function at the origin. After calculating Taylor polynomials of several rational functions in this way, and graphing the functions, polynomials and remainders, students are prepared to understand Taylor polynomial approximations for general functions and Lagrange's formula for the remainder.

Michael J. Seery, Pursuit and Regular n-gons
Generalizing a well-known problem, the author considers the paths followed by persons at the vertices of a regular n-gon, each of whom walks at all times at unit speed toward the person located k vertices away. A differential equation satisfied by the pursuit spirals is found and a plot of sample paths suggests several exercises for calculus students.

Computer Corner

Anne M. Burns, Modeling Trees with a Stochastic Matrix
The geometry of a binary tree can be described by a lower triangular stochastic matrix, the ramification matrix of the tree. Conversely, given such a matrix, by interpreting its entries as probabilities one can generate collections of random trees with visually similar branching behaviors. By adjusting the probabilities in the matrix we can tailor the model to produce trees with a variety of realistic shapes. Suggestions follow on extensions for modeling natural branching processes with computer graphics.

Software Review

Standard Math Interactive is a compact disk version of the CRC Standard Mathematical Tables and Formulae that makes possible quick computer searches of the contents of this large reference handbook. Once a mathematical expression is found, operations on it such as evaluation, plotting, integration, or expansion in series can be selected from menus and performed by a special version of the computer algebra system Maple V that is an integral part of Standard Math Interactive.