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A Panoply of Polygons

By Claudia Alsina and Roger B. Nelsen. DOL/58, 2023, 267 pp.

Are you a polygonophile? If not, A Panoply of Polygons will make you one. The book begins with basic facts about polygons and the tools that are helpful for understanding them. Chapter 2 is devoted to pentagons, Chapter 3 to hexagons, Chapter 4 to heptagons. See the pattern? Since it is a basic requirement of any book that it be finite, Chapter 6 treats “many-sided polygons.” Miscellaneous classes of polygons are presented in Chapter 7, figures such as centrally symmetric convex polygons called “zonogons” (from an ancient Greek word for belt or girdle). Combinatorics, which makes guest appearances throughout the show, gets a star role in the final chapter, “Polygonal Numbers.” The reader will enjoy many theorems about polygons and related figures proved with only a few words (or no words at all). The authors pack an impressive amount of mathematics and history into only 267 rectangular pages.

While many polygons are impossible to construct with straightedge and compass, Major League Baseball’s home plate, described in the League’s handbook, is impossible to construct with any tools in our world. (The problem is the right angle in front of the batter.) I doubt that many readers already knew this, or that the 1000 kwacha coin from Zambia has heptagonal shape. Factoids such as these appear with illustrations throughout A Panoply of Polygons. They are sprinkled spice, but they never overwhelm the theorems and history that are the heart of the book.

A Panoply of Polygons offers a panoply of reasons to be fascinated by its subject. If you love geometry, then this book will delight you. If you have forgotten how beautiful geometry is, then this book will remind you.

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Integer and Polynomial Algebra

By Kenneth R. Davidson and Matthew Satriano. MAWRLD/31, 2023, 185 pp.

This small book resulted from a course taught for many years at the University of Waterloo, Canada. The goal of the course — and of the book — is to introduce students to rigorous mathematics using the familiar setup of numbers and polynomials.

The first five chapters are about numbers. Chapter 1 is about integers with the main focus on the most important notions and results: the Euclidean algorithm, unique factorization into primes, and the infinity of the number of primes. Chapter 2, which is about modular arithmetic, explains the Chinese remainder theorem and Fermat’s little theorem. Chapter 3 discusses quadratic extensions, explaining, in particular, that integers in some quadratic extensions do not have the unique factorization property. Quadratic reciprocity is also discussed. Chapter 4 is about primality testing and the Rivest–Shamir–Adleman encryption algorithm, while Chapter 5 is a brief introduction to real and complex numbers, which culminates in the proof of the fundamental theorem of algebra.

The last two chapters of the book are about polynomials over the integers (unique factorization and the definition of algebraic numbers) and over the reals (Sturm algorithm of the determination of the number of real roots). Finally, Chapter 7 presents the basics of the theory of finite fields, treating them as quotient rings of the ring of polynomials in one variable over a prime field.

Attractive features of the book are its wealth of exercises (5–10 for each of 50+ sections) and brief notes at the end of each chapter suggesting further reading. With these features, instructors will have many options to tailor a math circle activity, a summer camp program, or an undergraduate course in a way most appropriate for their students.