Much of this issue of the Magazine demonstrates the connections between different areas of mathematics. Barnes, Burry, Russell, and Schaubroeck visually examine Julia sets that emerge when complex functions are iterated. Brown offers more names for the (7,3,1) design through its connections to codes, finite geometries, difference sets, and real normed algebras. Berele and Catoiu relate the rationalizing of denominators to symmetric function theory, field theory, and algebraic number theory. Swart and Shelton generalize a card trick from The Ellen DeGeneres Show. Byer and Smeltzer extend a geometric result about three mutually tangents circles in the plane to higher dimensions. The issue rounds out with two proofs without words, a crossword puzzle, Reviews and Problems. —Michael A. Jones, Editor
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Vol. 88, No. 2, pp 90 – 160
Articles
Emerging Julia Sets
Julia A. Barnes, Clinton P. Curry, Elizabeth D. Russell, and Lisbeth E. Schaubroeck
Functions from the complex plane to itself are difficult to visualize; we consider the real and imaginary projections. In this paper, we explore the connections between the graphs of the real and imaginary parts of various complex functions and their corresponding filled Julia sets. We begin by examining the family of complex quadratic functions.We then expand our results to a broader collection of rational maps, including functions whose Julia sets form a Cantor set of simple closed curves, checkerboards, and a perturbed rat.
To purchase from JSTOR: 10.4169/math.mag.88.2.91
Many More Names of (7, 3, 1)
Ezra Brown
The (7, 3, 1) block design is an object that shows up in many areas of mathematics. In fact, (7, 3, 1) seems to appear again and again in unexpected places. A 2002 paper described (7, 3, 1)'s connection with such areas as graph theory, number theory, topology, round-robin tournaments, and algebraic number fields. In this paper, we show how (7, 3, 1) makes appearances in the areas of error-correcting codes, n-dimensional finite projective geometries, difference sets, normed algebras, and the three-circle Venn diagram.
To purchase from JSTOR: 10.4169/math.mag.88.2.103
Rationalizing Denominators
Allan Berele and Stefan Catoiu
We present several techniques for rationalizing the denominators of fractions which involve radical expressions of rational numbers. Our algorithms are based on prime numbers, indeterminate coefficients, symmetric polynomials, and Galois theory.
To purchase from JSTOR: 10.4169/math.mag.88.2.121
Notes
Revelations and Generalizations of the Nine Card Problem
Breeanne Baker Swart and Brittany Shelton
The nine card problem is a magic trick performed by shuffling nine playing cards according to a set of rules. The magic is that a particular card will always reappear. The success of this trick can be easily explained by considering the lengths of the words in the names of playing cards, which define the shuffling rules. In this paper, we use permutations to prove that the trick will always work. We then use this methodology to generalize the trick to any number of cards with shuffles according to different rules
To purchase from JSTOR: 10.4169/math.mag.88.2.137
Proof Without Words: President Garfield and the Cauchy-Schwarz Inequality
Claudi Alsina and Roger B. Nelsen
We prove wordlessly the Cauchy–Schwarz inequality (for n = 2) using a trapezoid partitioned into three right triangles.
To purchase from JSTOR: 10.4169/math.mag.88.2.144
Mutually Tangent Spheres in n-Space
Owen D. Byer and Deirdre L. Smeltzer
In this note we prove that the points of tangency of n + 1 mutually tangent spheres in n-dimensional space lie on a generalized sphere. Coxeter's observation that for each of five mutually tangent spheres there is a sphere passing through the six points of mutual contact of the remaining four is a corollary of this result in the n = 3 case.
To purchase from JSTOR: 10.4169/math.mag.88.2.146
Proof Without Words: A Trigonometric Identity for sec x + tan x
Rober B. Nelsen
We prove wordlessly the identity sec x + tan x = tan(π/4 + x/2).
To purchase from JSTOR: 10.4169/math.mag.88.2.151
Unifying Two Proofs of Fermat's Little Theorem
Massimo Galuzzi
A new simple proof of Fermat's little theorem is given that generalizes the proofs given in this MAGAZINE by Levine (1999) and Iga (2003).
To purchase from JSTOR: 10.4169/math.mag.88.2.152
Books for a Math Audience
Brendan W. Sullivan
To purchase from JSTOR: 10.4169/math.mag.88.2.155
Problems and Solutions
Proposals, 1966-1970
Quickies, 1049-1050
Answers, 1049-1050
To purchase from JSTOR: 10.4169/math.mag.88.1.72
Reviews
Why do you do mathematics?; twin primes on film; constitutional mathematics; fun!
To purchase from JSTOR: 10.4169/math.mag.88.2.159