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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**