Configuration Spaces of Planar Pentagons
By Robyn Curtis and Marcel Steiner
The configuration space of a pentagon Π is defined as the topological space of all possible realizations of Π in the plane R², modulo the group of orientation-preserving isometries of R². By considering the pentagon as a mechanical linkage, the configuration spaces of all planar pentagons can be classified using techniques from basic topology. There are twenty-one topologically distinct spaces: the orientable surfaces of genus 0, 1, 2, 3, and 4 with up to 4, 3, 2, 1, and 0 singularities, respectively; the union of two tori identified at up to three points; a set consisting of one point; the empty set. We introduce an alternative method for determining the configuration space using an extended version of the Euler characteristic. The centerpiece of this work is a formula for that makes it possible to classify the configuration space of any given pentagon within a matter of seconds.
Circles in Circles: Creating a Mathematical Model of Surface Water Waves
By Katherine Socha
During the month of June in the year 1943, an Italian supply ship had the misfortune to encounter a British submarine, the United. Soon, a passing airplane joined the fray. The resulting air-sea battle was photographed by a RAF fighter pilot on reconnaissance. This photograph, now legendary in fluid dynamics circles, forms the basis for a problem in mathematical modeling of surface wave motion. How can we account for the qualitative difference between the motion of so-called gravity waves, as created by the sea battle, and the motion of so-called capillary waves, as created by raindrops striking a pond? What difference does the scale make?
Conway's Napkin Problem
By Anders Claesson and T. Kyle Petersen
The napkin problem was first posed by John H. Conway, and written up as a 'toughie' in Mathematical Puzzles: A Connoisseur's Collection, by Peter Winkler. To paraphrase Winkler's book, there is a banquet dinner to be served at a mathematics conference. At a particular table, n men are to be seated around a circular table. There are n napkins, exactly one between each of the place settings. Being doubly cursed as both men and mathematicians, they are all assumed to be ignorant of table etiquette. The men come to sit at the table one at a time and in random order. When a guest sits down, he will prefer the left napkin with probability p and the right napkin with probability q = 1 - p. If there are napkins on both sides of the place setting, he will choose the napkin he prefers. If he finds only one napkin available, he will take that napkin (though it may not be the napkin he wants). The third possibility is that no napkin is available, and the unfortunate guest is faced with the prospect of going through dinner without any napkin! Using a combinatorial approach, we answer questions like: What is the probability that every guest receives a napkin? How many guests do we expect to be without a napkin? How many guests are happy with the napkin they receive?
A Simple Example of a New Class of Landen Transformations
By Dante Manna and Victor H. Moll
The classical arithmetic-geometric mean discovered by Gauss is based on a transformation of the parameters of an elliptic integral that leaves the integral invariant. The authors present the simplest example of such transformations in the case of the integral of a rational function. We present an explicit algorithm that leaves the integral invariant and establish the convergence of its iterates.
Exactly Which Bounded Darboux Functions are Derivatives?
By Michael W. Botsko
Riemann Mapping Theorem by Steepest Descent
By Mark Levi
Entire Functions that Tend to Zero on Every Line
By David H. Armitage
Stirling's Formula Via the Poisson Distribution
By Mark A. Pinsky
Problems and Solutions
God Created Integers
Edited, with commentary, by Stephen Hawking
Reviewed by John Stillwell