The Mathematical Association of America has selected the winners of the 2012 Beckenbach Book Prize.

Full citations and biographical information for each winner is available below.

The Beckenbach Book Prize, established in 1986, is the successor to the MAA Book Prize which was established in 1982. It is named for the late Edwin Beckenbach, a long-time leader in the publications program of the Association and a well-known professor of mathematics at the University of California at Los Angeles. This prize is awarded to an author of a distinguished, innovative book published by the MAA. The award is not given on a regularly scheduled basis, but is given only when a book appears that is judged to be truly outstanding.

This award will be presented during the Joint Prize Session on Thursday, January 5, 2012, at the 2012 Joint Mathematics Meetings in Boston, Massachusetts.

Just how interesting can a book be whose only topic is polynomials? It doesn't take long to find out. By page 3, we encounter Lill's ingenious paper-folding technique for visualizing real roots of a polynomial, with an elegant proof based on Horner's method to evaluate a polynomial. Now most mathematicians are vaguely aware of Horner's method, but try to find one who knows Lill's technique, or one that knows that the first mention of Lill's idea in the United States appeared in an 1879 pamphlet by a lieutenant in the Army Corps of Engineers. By Chapter 2 we find that if a cubic *p*(*x*) has three noncollinear roots in the complex plane, the unique ellipse inscribed in the triangle formed by those roots has the roots of *p*Ê¹(*x*) as it foci (Marden's Theorem), and that a polynomial *p*(*x*) with positive integer coeffcients is completely determined by the value of *p*(1) = *b* and *p*(*b*) (think base *b*). There is much, much more: palindromials, polynomial interpolation, symmetric functions, Newton's identities, and a brief history of Cardano's formula for the roots of a cubic (again, who knew Mark Kac sharpened his teeth at age 15 on his own method of getting Cardano's formula?).

And that is only the first third of the book! The next part covers max/min problems, beginning with a careful analysis of the Lagrange fallacy then moving on to the Milkmaid problem, rotating ellipses, the ladder-around-the-corner (done with envelopes!), and the old shortest-time path through-or-around a field but with a discontinuous objective function. The last section focuses more on calculus questions, the wonder of elementary functions, and the expansion of the toolbox with special functions to solve, for example,* ex = cx*.

The exposition is perfect: expansive, relaxed, and detailed (almost nothing is stated without being proved). Although knowledge of calculus is needed, any good undergraduate mathematics major can read the entire book, as could many talented high school students. The book is a goldmine of topics for undergraduate talks.

*Biographical Note*

Dan Kalman has been a member of the mathematics faculty at American University, Washington, DC, since 1993. Prior to that he worked for eight years in the aerospace industry and taught at the University of Wisconsin, Green Bay. Kalman has a B.S. from Harvey Mudd College and a Ph.D. from University of Wisconsin, Madison. He has been a frequent contributor to all of the MAA journals, and has served on the editorial boards of both MAA book series and journals; "Uncommon Mathematical Excursions" is his second MAA book. Kalman has served the national and regional MAA in several capacities, including a term as associate executive director for programs, as the current governor for the MD-DC-VA section, and as a cast member of MathFest 2011's *MAA - the Musical*. While he delights in word play of all kinds, his propensity for making puns has been grossly exaggerated.

Burnside's classic text, *Theory of Groups of Finite Order*, was the first book written on group theory; it begins by defining a group as a collection of operators closed under multiplication and inverses (no associative law). Groups don't just sit there, they do something, they act on something, as permutations or symmetries or isometries. This is where groups come from, and it is the standard viewpoint both of users in physics, geometry, analysis, topology, combinatorics, and also of specialists in permutation groups, representations, and geometric group theory. Unfortunately, this viewpoint is almost absent from present undergraduate textbooks.

Nathan Carter's eye-opening textbook has a mission to fix that. Using a Rubik's Cube as a motivating example, he returns to Burnside's definition, only with ’actions’ instead of ’operators’. This is followed by Cayley diagrams to show how a group looks; Dehn's term *gruppenbilder*, or group pictures, was always the better name. Next come motivating examples: symmetries of polygons, crystals, frieze patterns, contra dancing, roots of polynomials. Chapter 4, entitled "Algebra at Last," finishes with the ’classic’ definition of a group, although the action definition is really the classic one. Lastly, much of the traditional content of an undergraduate text on group theory can be found in Carter’s text: the alternating and symmetric groups, abelian groups, subgroups, products and quotients, homomorphisms, Sylow theory, and a little Galois theory.

The presentation is never traditional. Turn to any page and a figure jumps out: Cayley diagrams, colored multiplication tables, permutations as arrows from letter to letter - even Sylow theory is accompanied by conceptual sketches outlining the proofs. The exposition is breezy and leisurely: theorems are "taken out for a test drive," a proposition "gives us a headstart," a statement is "dense with notation" or "hairy." Steps are explained, proof strategies are analyzed - anything to loosen the seductive (after the fact) but intimidating (before the fact) concision and formality of algebra. The result is a textbook that reads like a conversation. Nathan Carter's *Visual Group Theory* is an original breakthrough that has a chance of transforming a staple of a mathematics major's diet.

*Biographical Note*

Nathan Carter uses computer science to advance mathematics. He studied both subjects at the University of Scranton and at Indiana University, earning a Ph.D. in mathematical logic in 2004. Besides work in logic, he has written a book on group theory visualization, and done a little work in social network analysis. His open source mathematics software, including packages for group theory visualization, games in formal logic, and a general validation environment for mathematical reasoning, is available at http://www.platosheaven.com.