American Mathematical Monthly -March 2000

March 2000

Yueh-Gin Gung and Dr. Charles Y. Hu Award for Distinguished Sevice to Paul R. Halmos
by Andrew M. Gleason
gleason@abel.math.harvard.edu

Tiling with Squares and Anti-Squares
by Chris Freiling, Robert Hunter, Cynthia Turner, and Russell C. Wheeler
cfreilin@csusb.edu, Rlsmj5@aol.com, cturner@math.csusb.edu, rcwheel@juno.com
A famous old result of Max Dehn from 1902 says that a rectangle can be tiled by squares if and only if the ratio of its sides is a rational number. Actually, tiling such a rectangle is easy; the hard part is to show that rectangles with irrational ratio can't be tiled. But what if we try to answer the same question for L-shapes? An L-shape is a six-sided rectilinear polygon. At first glance, it seems as if the answer should be a direct corollary of Dehn's work, or at least an easy corollary of one of the many subsequent proofs of Dehn's theorem. But that doesn't seem to be the case and, to our knowledge, the problem is still open! In this paper we discuss some techniques that might be useful for attacking this problem. We do this by solving a much easier problem: Which rectilinear polygons can be tiled if we use both squares and anti-squares? An anti-square acts like a square that is made out of anti-matter, so that a square and anti-square annihilate each other in the region of overlap. Actually, this time the easy part is to show that certain regions cannot be tiled. This is the easy part because we just modify a very clever proof of Dehn's result given by Swiss geometer, Hadwiger. Most of our effort is in showing how to tile the other regions, the ones for which the impossibility proof doesn't work.

Solving Linear Differential-like Equations
by Luis Verde-Star
verde@xanum.uam.mx
We present a method for the solution of linear functional equations involving an operator that behaves like differentiation in a certain linear algebraic sense. Such equations include a large class of linear difference and differential equations with constant or variable coefficients.

The basic ideas are presented first in the case of linear differential equations with constant coefficients. One-sided inverses of differential operators are obtained by means of a convolution product that is defined algebraically on the vector space of exponential polynomials. We show that the same constructions can be done in an abstract linear space. We use only basic concepts and results from linear algebra and some properties of polynomials and rational functions.

Special Relativity with Acceleration
by Garry Helzer
gah@math.umd.edu
Relativity theory has been around for nearly a century but no mention of it can be found in calculus or elementary differential equations courses. Consider the standard problem of a ball thrown into the air. How does the relativistic behavior differ from the Newtonian? A question like this is thought to be unapproachable on an elementary level because, first of all, problems involving acceleration require the general theory of relativity and, second, even the special theory requires Lorentz transformations and these have no natural place in the standard courses. Both ideas are wrong. Problems like the thrown ball are discussed using standard concepts from calculus and elementary differential equations.

Lattice Polygons and the Number 12
by Bjorn Poonen and Fernando Rodriguez-Villegas
poonen@math.berkeley.edu, villegas@math.utexas.edu
In this article, we discuss a theorem about polygons in the plane, which involves in an intriguing manner the number 12. The statement of the theorem is completely elementary, but its proofs display a surprisingly rich variety of method, and at least some of them suggest connections between branches of mathematics that on the surface appear to have little to do with one another.

Notes

Rhombic Penrose Tilings Can Be 3-Colored
by Tom Sibley and Stan Wagon
tsibley@csbju.edu, wagon@malacaster.edu

Cubic Fields and Radical Extensions
by Ming-chang Kang
kang@math.ntu.edu.tw

Generating the Measures of n-Balls
by Lee Badger
lbadger@weber.edu

The Volume of the Unit Ball in Cⁿ
by Omar Hijab
hijab@math.temple.edu

Partial Fractions and Four Classical Theorems of Number Theory
by Michael D. Hirschhorn
m.hirschhorn@unsw.edu.au

Smooth (C^x) but Nowhere Analytic Functions
by Sung S. Kim and Kil H. Kwon
sskim@newton.hanyang.ac.kr, khkwon@jacobi.kaist.ac.kr

The Evolution of...
Comments on the Paper "Two Letters by N. N. Luzin to M. Ya. Vygodskii"
by Detlef Laugwitz

Problems and Solutions

Reviews

Graphica 1. The Imaginary Made Real: The Images of Michael Trott
Graphica 2. The Pattern of Beauty: The Art of Igor Bakshee.

Reviewed by Harold P. Boas
boas@math.tamu.edu