American Mathematical Monthly - October 2001

OCTOBER 2002

Inside the Lévy Dragon
by Scott Bailey, Theodore Kim, and Robert S. Strichartz
smb43@cornell.edu, kim@cs.unc.edu, str@math.cornell.edu The Lévy Dragon is a well-known fractal introduced by Paul Lévy in 1938. It is a connected subset of the plane with interior (in fact it tiles the plane) but the interior is disconnected. Although the dragon has a fractal boundary of dimension 1.934007Â…, we show that each component of the interior has a polygonal boundary (with perhaps infinitely many edges) of finite length. There are infinitely many components, but we conjecture that they are all similar to one of sixteen different shapes. We show pictures of these shapes and some of the ways they interweave when two smaller dragons combine to make a larger dragon. We explain how we used the computer as a kind of "microscope" to reveal this structure. More pictures and programs are available on the web site http://www.mathlab.cornell.edu/~twk6/.

The Orthic Triangle and the OK Quadrilateral
by Anthony Philippakis
Anthony_philippakis@hotmail.com It is a maxim of classical geometry that the shortest distance between any two points is a straight line. Using this principle as our guiding intuition, we investigate the geometry of inscribed paths of minimal perimeter in polygons and polyhedrons. Our central technique is to view lines and planes not only as geometric object, but as the defining data for reflections of the plane and space. We show that the desired path is merely a straight line that has been reflected off each of the faces. We take as examples the planar and elliptic acute triangles, and the acute tetrahedron. In each case, an explicit construction is given for the desired polygon of minimum perimeter.

Blet: A Mathematical Puzzle
by F. Rodriguez Villegas, L. Sadun, and J.F. Voloch
villegas@math.utexas.edu, sadun@math.utexas.edu, voloch@math.utexas.edu Blet is a puzzle. We describe it and present its solution. In the process we discuss some aspects of the group of two-by-two integer matrices, simulated annealing, and some combinatorics. We end by posing some problems related to Blet.

Kemeny's Constant and the Random Surfer
by Mark Levene and George Loizou
mark@dcs.bbk.ac.uk, george@dcs.bbk.ac.uk We revisit Kemeny's constant in the context of Web navigation, also known as "surfing." We generalize the constant, derive upper and lower bounds on it, and give it a novel interpretation in terms of the number of links a random surfer will follow to reach his final destination.

Problems and Solutions

Notes

Partial Fractions, Binomial Coefficients, and the Integral of an Odd Power of Sec
by Daniel J. Velleman
djvelleman@amherst.edu

A Solution to the Bent Wire Problem
by Per E. Manne and Steven R. Finch
per.manne@nhh.no, sfinch@mathsoft.com

A Probabilistic Derivation of Dirichlet Integrals
by B. Liseo
brunero.liseo@uniroma1.it

A Note on a Closed Formula for Poly-Bernoulli Numbers
by Roberto Sánchez-Peregrino
sanchez@math.unipd.it

An Elementary Calculation of the Dihedral Angle of the Regular n-Simplex
by Harold R. Parks and Dean C. Wills
parks@math.orst.edu, dwills@math.orst.edu

Reviews

Mathematical Expeditions: Chronicles by the Explorers by Reinhard Laubenbacher and David Pengelley
Reviewed by Arthur M. Hobbs
hobbs@math.tamu.edu

The Lebesgue-Stieltjes integral: a Practical Introduction
by Michael Carter and Bruce van Brunt
Reviewed by Robert G. Bartle
RGBartle@aol.com

Editor's Endnotes