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Uncommon Mathematical Excursions: Polynomia and Related Realms

Dan Kalman
Mathematical Association of America
Publication Date: 
Number of Pages: 
Dolciani Mathematical Expositions 35
BLL Rating: 

The Basic Library List Committee suggests that undergraduate mathematics libraries consider this book for acquisition.

[Reviewed by
Debbie Gochenaur
, on

Offering an assortment of topics in the algebra, geometry, and calculus curricula, this book is intended as an enrichment for those familiar with these topics at the upper-secondary or introductory college math level. While the book may be geared especially for teachers who have taught their courses enough times to be thoroughly comfortable with the content, advanced students, as well as scientists and mathematicians in general, may find topics within this book intriguing. It has the potential to appeal to a broad audience. The book is meant to guide the readers in exploring ideas that are related to his/her core mathematical knowledge.

The book is divided into three parts — The Province of Polynomia, Maxiministan, and The Calculusian Republic. Kalman ensures that the reader can make connections to known undergraduate mathematics and advanced secondary topics in the algebra, geometry, and calculus domains; he works to help the reader make extensions and perhaps understand more clearly the depth of mathematics in these seemingly elementary topics. Overall, the reader may be surprised by aspects of a particular topic that will lead to a greater understanding and appreciation for the mathematics.

This book is not meant to be a textbook — it is a journey linking new ideas to familiar ones. Some topics are meant to be lingered over, taking time to ponder the concepts, with more details and ideas provided. In the Province of Polynomia — solving polynomial equations including alternate solutions for cubics and quartics together with the historical reference for when each was discovered. In Maxministan — the hallway problem which deals with moving a ladder horizontally around a corner, leading to envelopes, includes an extension as well as a discussion of duality. In The Calculusian Republic — envelopes including boundary points, intersections and asymptotes.

Other topics are meant to be quick day jaunts into a variety of areas, each self-contained; a number of these shorter trips are included in the miscellaneous chapters in each part of the book. These include palindromials, Marden’s Theorem, borders on string art where boundaries appear to be smooth curves but are in reality polygonal paths, and isoperimetric duality. While some topics may be familiar to the reader, everyone is sure to find something interesting here.

The book includes descriptions of the concepts, setting them in historical context, and a variety of examples with some proofs. For example, the discussion on solutions for quartics begins with Descartes’ 1637 factoring of the quartic into two quadratics. With algebraic manipulation, the resulting equation is a sixth degree polynomial that is a cubic equation in u2, which Kalman has discussed in previous sections. After commenting about the similarity of Descartes’ method to Ferrari’s, Kalman moves on to Euler’s 1770 solution to the quartic, linking it to Euler’s work on the cubic. Beginning by assuming that x is the sum of the square roots of r, s, and t, x is squared, simplified, and then squared again. Kalman explains that equating coefficients and substituting enables Euler to rewrite this system so that you finally find that the original r, s, and t are the roots of a cubic polynomial.

A third algebraic approach to solving quartics is given before Kalman moves on to explain the connections between solving quartics and cubics as well as Lagrange’s frustrated but foundational struggle to extend this work to quintics. There are sidebars throughout the book, including one on Lagrange and Vandermonde with a brief mention of their work with polynomial equations as well as giving a broader picture of their contributions, placing Lagrange as the greatest mathematician between Euler and Gauss’ generations. Every chapter ends with a brief history of the topics covered, with references and additional print sources if the reader wishes to explore further.

As a supplementary resource to the book, Kalman has created a companion website that includes a quick review of the background mathematics necessary to understand the material. For algebra, this includes a discussion of polynomial division, remainder and factor theorems, complex numbers, and n-th roots of unity; the section on matrices is also complete while the multivariable calculus review is still in the works. There is a page of links for electronic sources from the Bibliography as well as several animations to illustrate some of the content discussed in the book. The animations are accompanied by clear explanations of not only how to manipulate the animations but also the meaning of the mathematical content.

The only drawback to this book is that there are no exercises with which the reader can apply his/her new knowledge; supplementary exercises are still being developed for the companion website and will only be available for some of the chapters. It seems that the website is meant to be an integral part of the reader’s experience; if so, perhaps the publication of this book should have been postponed until the website was completed.

This book would make great summer reading or provide a good source for individual or group project assignments. Kalman integrates the metaphor of traveling and exploring the back country of mathematics as he introduces the reader to the Province of Polynomia, Maxministan, and then on to The Calculusian Republic. Kalman’s sense of humor and adventure comes through from cover to cover while he takes you on the roads less traveled making connections to the mathematics you love.

Debbie Gochenaur is Assistant Professor of Mathematics at Elizabethtown College. Her interests lie in math learning disabilities and encouraging underrepresented minorities in STEM. 

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