The Shape of Algebra in the Mirrors of Mathematics

Technology can significantly facilitate mathematical exploration. Little was achieved by plotting points on graph paper when compared to the 3D graphs possible with a few keystrokes and the right software. The Shape of Algebra in the Mirrors of Mathematics uses software to explore algebraic equations geometrically, leading naturally to topics (introductions only) in algebra, algebraic geometry, topology, complex variables, and algebraic topology. A disk bundled with the text contains the VisuMatica software, a 34 page manual explaining its use, and color pictures of all figures. VisuMatica is a tool that can develop intuition and facilitate creativity. Using it gives one the chance to think like a mathematician discovering results, in contrast to most texts that show how to think like a mathematician proving results already discovered.

The authors hope to make ideas usually introduced at the graduate level available to undergraduates. Their approach is novel and the 600-page text contains a wealth of material that may lend itself to a variety of instructional uses. It is well worth investigating especially if one takes the time to get familiar with VisuMatica. Unfortunately there is no index, which is an annoyance, and there are some misprints.

Rather than look at one object, the authors look at families of objects and mappings. To introduce the software, the first chapter treats families of rectangles, using two representations: length-height and area-perimeter, with mapping from one space of rectangles to the other. Many exercises suggest using VisuMatica to explore the transformation from one representation to the other. Each chapter ends with a section "How to use VisuMatica in this chapter" which, referring to tags in the exercises, shows how to use the tool.

The second chapter explores the mapping of quadratic equations from the root representation to the coefficient one in real numbers. Chapter 3 introduces complex numbers and Chapter 4 looks at the geometry of complex linear polynomial maps, defining a group, subgroup, and homomorphism. Chapter 5 starts the material that many may not have seen. Beginning to look at complex polynomial maps and closed plane curves leads to the investigation of loops. This chapter includes rope and train models but also definitions of a topological space, homotopic maps, the fundamental group, and proofs of some theorems. One could target this material to vastly different audiences depending on how much of the advanced material is included.

The next chapters cover the geometry of complex polynomial maps of degree two and three followed by cubic and quartic polynomials, including solutions by radicals and Riemann surfaces. The book concludes with chapters on complex maps of high degree and the fundamental theorem of algebra, and polynomial maps and singular points.

VisuMatica is a wonderful tool with which to explore. This book has many thought-provoking exercises with which to build insight. It is uneven in level, with definitions popping up here and there and a few theorems too. More editing and certainly an index would have helped, but it is well worth looking at to get into the discovery mode.

Art Gittleman (artg@csulb.edu) is Professor of Computer Science at California State University Long Beach.