# American Mathematical Monthly - October 2003

## OCTOBER 2003

Richard Brauer: Sketches from His Life and Work
by Charles W. Curtis
cwc@darkwing.uoregon.edu
Richard Brauer's research exemplified twentieth century mathematics at its best. In this biographical essay, I will try to give an impression of him as a person, beginning with his student days at the University of Berlin. I plan to give an account of some of his interactions with people who contributed to his development as a mathematician and with those who collaborated with him, including Helmut Hasse and Emmy Noether while he was in Germany, and Hermann Weyl, Cecil Nesbitt, and Tadasi Nakayama after he moved to North America. I have also included sketches of how his work progressed at different periods of his life, and what it means for us today.

Altitudes of a Tetrahedron and Traceless Quadratic Forms
by Hans Havlicek and Gunter Weiss
havlicek@geometrie.tuwien.ac.at, weiss@math.tu-dresden.de
It is well known that the three altitudes of a triangle are concurrent at the so-called orthocenter of the triangle. So one might expect that the four altitudes of a tetrahedron also meet at a point. However, it was already pointed out in 1827 by the Swiss geometer Jakob Steiner (1796-1863) that the altitudes of a general tetrahedron are mutually skew. In this article we want to visualize some of the classical results on a tetrahedron and its altitudes. Also, we aim at a modern coordinate-free presentation in terms of analytic geometry based on a Euclidean vector space. There is a natural link between tetrahedra and certain traceless quadratic forms that will lead us to an explicit expression for the equation of the quadric surface carrying the altitudes of a general tetrahedron.

Newton’s Rule of Signs for Imaginary Roots
by Daniel J. Acosta
dacosta@selu.edu
We introduce a rule of Sir Isaac Newton that provides a lower bound for the number of imaginary (complex but not real) roots of a polynomial. J.J. Sylvester produced the first satisfactory proof of this rule, almost two hundred years after Newton's formulation.

The Number of "Magic" Squares, Cubes, and Hypercubes
by Matthias Beck, Moshe Cohen, Jessica Cuomo, and Paul Gribelyuk
matthias@math.binghamton.edu, bj91859@binghamton.edu, jessica@math.Binghamton.edu, pavel@math.utexas.edu
We define a magic square to be a square matrix whose entries are nonnegative integers and whose rows, columns, and main diagonals sum up to the same number. We prove structural results for the number of such squares as a function of the size of the matrix and the line sum. We give examples for small sizes and show similar results for symmetric, pandiagonal magic squares, and magic hypercubes.

The Sixty-third William Lowell Putnam Mathematical Competition
by Leonard F. Klosinski, Gerald L. Alexanderson, and Loren C. Larson

Notes

Erdös-Mordell Type Inequalities in a Triangle
by Razvan A. Satnoianu
r.a.satnoianu@city.ac.uk

Stirling’s Series Made Easy
by Chris Impens
ci@cage.rug.ac.be

A New Proof of Wedderburn’s Theorem
by Nicolas Lichiardopol
lichiar@club-internet.fr

After the Determinants are Down: A Criterion for Invertibility
by Karim Boulabiar and Gerard Buskes
karim.boulabiar@ipest.rnu.tn, mmbuskes@hilbert.math.olemiss.edu

Problems and Solutions

Reviews

It Must Be Beautiful: Great Equations of Modern Science
Edited by Graham Farmelo
Reviewed by Alex Kasman
kasmana@cofc.edu

Vector Calculus, Linear Algebra, and Differential Forms: a Unified Approach (2nd edition).
by John H. Hubbard and Barbara Burke Hubbard
Reviewed by Warwick Tucker
warwick@math.uu.se