Supplement to Means Appearing in Geometric Figures, by Howard Eves in the October 2003 issue of the Magazine: An animated demonstration (in Geometer's Sketchpad) of the means in a trapezoid, by Shannon Umberger Patton.
A Julia Set Is Everything
Julia A. Barnes and Lorelei M. Koss
A Julia set, the part of the domain where a complex function behaves chaotically, is often thought of as an object with an intricate and beautiful picture. Many Julia sets look like they have area zero and are surrounded by a large region where the function has very predictable behavior. This paper describes a particular complex rational function that has a Julia set equal to its entire domain. The picture of such a Julia set is quite dull compared to other examples, but the function has very interesting dynamics because it is chaotic everywhere.
An Algorithm to Solve the Frobenius Problem
Robert W. Owens
Given three positive integers, what is the largest integer that cannot be expressed as a nonnegative linear combination of those three integers? This is a special case of the Frobenius problem, which asks the same question for n positive integers with greatest common divisor 1. If we had three integers instead of two, a simple polynomial yields the answer, but with three or more integers, no such solution is possible except in a few special cases. So solutions must be constructed via some algorithm rather than being obtained by the evaluation of a function in closed form. A theorem characterizing of the solution of the general Frobenius problem leads to a brute force, exhaustion algorithm. This paper presents another algorithm to solve the Frobenius problem in general. The algorithm extends an older algorithm that held only for combining three integers.
Leibniz, the Yijing, and the Religious Conversion of the Chinese
Frank J. Swetz
Gottfried Wilhelm Leibniz was a synthesizer of ideas, a deep religious thinker and and an avid Sinophile. In his quest to order all human knowledge, he sought the characterista universalis or universal language that would assist this task. In his search, he was drawn to the Chinese Yijing or Book of Changes. Here in the concepts and symbols of the ancient classic, Leibniz found evidence to support his mystical model relating God to Creation, binary arithmetic. Through the Yijing he also became involved with the Figurist and accommodationist Jesuit strategies of converting the Chinese people to Christianity. This essay examines this strange episode in the history of mathematics and discussed how mathematics was linked with philosophy, cosmology, theology, and metaphysics to produce an enticing intellectual theory.
Means Appearing in Geometrical Figures
Howard W. Eves
Practically every student of mathematics is acquainted with the arithmetic and geometric means of two given positive numbers, a and b. Not so many students realize that there are many other means of two given positive numbers. A short list includes the arithmetic mean, the geometric mean, the harmonic mean, the heronian mean, the contraharmonic mean, the root-mean-square, and the centroidal mean. We give many examples where these means arise in geometric figures. In particular, each of the means in our list has a special interpretation as the lengths of slices of the trapezoid taken parallel to parallel sides of lengths a and b.
Mathematical Modeling for Metal Leaves
Yukio Kobayashi, Koichi Takahashi, Takashi Niitsu, and Setsuko Shimoida
Fractal geometry, a mathematical subject, has nonetheless become useful in many branches of science. This note introduces a graduation project on fractal geometry that draws on the areas of mathematics, computer science, and physical chemistry. Our paper compared the experimental method called electrodeposition with a simulation method called diffusion-limited aggregation (DLA). DLA is a simulation method that models various physicochemical phenomena: it is a simple random-growth process generated by Brownian motion. We use the mathematics of fractal geometry to compare these two methods. This mathematics allows us to see that a uniform distribution of numbers in the DLA simulation models this process more accurately than a normal distribution.
A Simpler Dense Proof Regarding the Abundancy Index
Richard F. Ryan
Let b represent a positive integer. The abundancy index of b is defined to be the sum of the factors of b, divided by b. The set of rational numbers that are abundancy indices is dense in the closed interval from one to infinity. The set of rational numbers, greater than one, which is not abundancy indices is dense in the same interval. A simple proof of the previous statement is presented in this note. A relationship between indices of the form (2n -1)/n and odd perfect numbers is also explored.
A Circle Stacking Theorem
In this article a Euclidean problem about stacking circles is explored, which came about from the consideration of the wine rack problem. The proof presented involves only the use of Euclidean transformations.
An Elementary Proof of Error Estimates for the Trapezoidal Rule
David Cruz-Uribe, SFO and C. J. Neugebauer
Essentially every calculus book contains the trapezoidal rule for estimating proper integrals, but most books omit the proof. Indeed, the standard proof is usually not accessible to first-year calculus students. We give an elementary proof which depends only on using integration by parts "backwards," and which should be understandable by most students. Our proof also generalizes to give estimates for functions that do not have bounded second or even first derivatives.
Triangles with Integer Sides
Michael D. Hirschhorn
We present a quick proof of the formula for the number of triangles with integer sides and perimeter n.
The Number of 2 by 2 Matrices over Z/pZ with Eigenvalues in the Same Field
This article involves 2 by 2 matrices over the finite field Z/pZ. The goal of the article is to find how many of them have eigenvalues in Z/pZ. The elementary part of the article involves the invertible upper triangle matrices. The more challenging part of the article uses the centralizer and conjugacy classes to prove a formula for the nonsingular matrices that form the general linear group of 2 by 2 matrices over Z/pZ.