A Bicentennial for the Fundamental Theorem of Algebra

Two hundred years ago, a promising young mathematician at the (later defunct) University of Helmstedt submitted a doctoral dissertation with the lengthy Latin title, Demonstratio nova theorematis omnem functionem algebraicam rationalem integram unius variabilis in factores reales primi vel secondi gradus resolvi posse: "A new proof of the theorem that every rational integral algebraic function in one variable can be resolved into real factors of first or second degree." Today, we might shorten things to "A new proof of the Fundamental Theorem of Algebra."

The student's name? Carl Friedrich Gauss.

From Figure to Form

Have you ever noticed the enchanting patterns in the wing of a butterfly, the fascinating symmetries in plants, and the fantastic shell constructions of snails and mussels? We find mathematics everywhere in nature: a veritable geometry text lying open right before our eyes.

Chess Queens and Maximum Unattacked Cells

There is now an enormous literature on the old classic task of placing eight queens on a chessboard so that no queen attacks another. There are twelve solutions, not counting trivial rotations and reflections. The task naturally generalizes to enumerating the number of solutions for n non-attacking queens on an n x n board.

Y2K: A personal history

Y2K. By now, everyone must be sick of hearing about it. The standard boilerplate about the Year 2000 computer problem runs as follows. In ancient times (the 1960s!), when computer memory was expensive, programmers eager to save scarce memory chose to represent dates with six digits, using two digits for the year (instead of four). It is said that these digital pioneers never expected their programs to survive into the enxt decade, much less the next centure. Moreover, the two-digit year problem--so the legend goes--is mainly confined to programs that were written for large mainframe computeres (like the old IBM 360 and 370 systems, for example) in banks, insurance companies, and other large businesses which are crucially concerned with keeping track of the date. But somehow, to the amazement of programming old-timers, their programs--now known as "legacy code"--have survived until the end of the century. And unless these old programs are repaired so they can handle dates in the new millennium, pandemonium will break loose upon the electronic foundations of western society.

The True Tale of a Tin Can

Mathematicians are rather odd creatures, some think. Their world view is a bit askew; their sensibilities slightly off center. Mathematicians tend to stare at the world in wonderment and find puzzles where others see merely the prosaic. They make their own fun. Once a mathematician settles upon a puzzle, he or she is apt to think about it, on and off, for some time. On occasion, the problem will be worked on actively. Often the puzzle will be allowed to percolate quietly in the subconscious. Frequently a mathematical model is made. Sometimes colleagues are consulted. Usually insight follows. On rare, wonderful occasions, the marvelous "Aha! experience" punctuates the process. In what follows, I present the true story of a real puzzle that exhibits all of these hallmarks.

Was Gauss Smart?

In my avocation as music director at St. Joseph the Worker Church in Hanson, Massachusetts, I often search for choral music both traditional and contemporary. The World Wide Web has turned out to be a terrific resource in this endeavor with many music publishers, composers, and music notation software companies providing links to sheet music or sound files.

Aliens, Asteroids, and Astronomical Odds

Many people have a fear of flying. For some it is merely an emotional stress, while others rearrange their lives repeatedly to avoid airplane travel. The fact that most of these individuals have heard that flying is safer than driving on a per mile basis does not seem to provide comfort.

Bridgework Or How I Became a Technical Writer

The motto of the Society of Technical Communication, the leading professional organization for technical writers, editors, and illustrators, claims that "Technical communicators are the bridge between the people who have ideas and the people who use them." It has always seemed a demeaning motto to me, implying as it does that technical communicators neither have ideas nor occasion to use them. A better motto might be, "Technical Communicators are highly skilled translators of jargon-ridden Geekspeak into clear, yet technically precise, English or some other standard language."

Problem Section:

S-35.

Proposed by R. S. Luthar, University of Wisconsin, Janesville. Prove that
cot A/2 + cot B/2 + cot C/2 > cot A + cot B + cot C
where A, B, C are angles of a triangle.

S-36.

Proposed by R. S. Luthar, University of Wisconsin, Janesville. Eliminate u and v from the following set of equations:
x = cos u + cos v, y = sin u - sin v, z = cos(u-v).

S-37.

Proposed by Mircea Ghita, Flushing, NY. Solve the equation
1/sin^{2k}(x) + 1/cos^{2k}(x)=8, where k is a positive integer.

S-38.

If d divides 2 n^2, prove that d + n^2 cannot be a perfect square.

The Final Exam: A Rose is a Rose is a Rose

(To read the winning entry to September's Final Exam, click here.)

Familiar to all of us is that sentimental refrain we giddily wont to waft to our beloved:

"(r=cos(4t)) are red, dear,
Violets are bluuuuueee--
Angels in heaven
Know I (r=1-sin(t)) youuuuu."

It's particularly fun to send this poem to someone who won't know what you're talking about, although the feedback is not guaranteed to be positive. Some people might think you're a little too flip with the sentiment and a little low on the versification. Then again, when you begin to explain the thrill of polar coordinates, people are apt to think you're fixating on the Titanic again. Well, we know where the real problem lies--it's not in the abstraction, it's in the equations. Whoever was fooled into thinking r=cos(4t) was a rose, and not a common daisy, anyway? We claim that what the world needs now is a better flower! To that end, we're offering up a competition to parametrize a flower (not necessarily a rose) that will be so elegant it will garner a Math Horizons t-shirt for up to 3 winners, and be reproduced in a future issue of Horizons. We just need that the flower be produced with real equations (no built-in clip art pictures, and furthermore, employees of Spielberg's Industrial Light and Magic need not apply). The flowers below were created by students in a Calculus class. Send your equations and a color print-out of your flower to Sandra Keith/Mathematics Dept./St. Cloud State U./ St. Cloud, MN 56304, by January 1, 2000 so a committee of judges (we like to think of ourselves as those refined ones who stop and smell the roses) can select a bouquet of winners.