Mathematica in action. Problem solving through visualization and computation. With CD-ROM. 3rd ed.

*(English)*Zbl 1198.65001
New York, NY: Springer (ISBN 978-0-387-75366-9/pbk; 978-0-387-75477-2/ebook). xi, 578 p. (2010).

This the third edition of the famous book “Mathematica in Action” in which the author guides beginner as well as veteran users alike through Mathematica’s powerful tools for mathematical exploration [For the second edition (2000) see Zbl 0944.65001]. The transition to Mathematica 7 is made smooth with a plenty of examples and case studies that utilize Mathematica’s newest tools such as dynamic manipulations and adaptive three-dimensional plotting. A brief summary of the twenty-one chapters of this book is as follows:

Chapter 0 contains notational conventions used in this book. An impressive number of data sets available in Mathematica is also described. Chapter 1 provides an introduction to the fundamental two-dimensional plotting functions of Mathematica. The chapter also includes an introduction to the generation of animations. Chapter 2 uses elementary number theory to introduce a variety of basic and advanced features of Mathematica. Several important techniques are introduced, as well as different approaches to graphing data. Chapter 3 presents an introduction to the two-dimensional graphic capabilities of Mathematica. As an example generating various images related to rolling wheels is focused. The final section presents several unusual applications of the wheel concept such as the generation of a curve on which a square can roll smoothly and a construction of a drill that makes perfect square holes. Chapter 4 introduces three-dimensional graphics. The basic tools are contour plots, density plots and three-dimensional surface plots.

The dynamic display features of Mathematica are covered in Chapter 5. The center piece of this chapter is the “Manipulate” command. The Cantor set is explored in Chapter 6 with extensions to the complex world. Chapter 7 shows how the in-built functions of Mathematica can be used to investigate the result of iterating functions. Chapter 8 shows how to simulate a turtle and use it to generate approximations of a space filling curve.

Mathematica has very powerful visualization tools which allow one to generate surfaces defined by two parameters. Chapter 9 contains several applications: exploring some properties of torus, generating a double torus and looking at some unusual surfaces such as the Costa surface. The chapter concludes with a surprising extension, due to Mandelbrot, of the Koch snowflake construction to three dimensions.

Tiling of the plane has both decorative and mathematical uses and has been studied for centuries. Chapter 10 gives a study of this. In Chapter 11 the author investigates fractal sets. Chapter 12 discusses the solving of equations. Mathematica has a variety of functions to be used in optimization problems. Chapter 13 contains several case studies on this topic. Differential equations are dealt with in Chapter 14. Chapter 15 shows how a published construction in computational geometry – the construction of a three-dimensional room that contains a point invisible to guards placed at every vertex – must be changed if it is to be correct. Chapter 16 explains how some built-in functions can be used to do group theory. Chapter 17 contains a discussion of the four-colour theorem that seeks to turn Kempe’s false proof of 1879 into a viable algorithm for four-colouring planar graphs.

The main theme of Chapter 18 is the calculation of \(\pi\) using new formulas. The Banach-Tarski paradox is discussed in Chapter 19. The Riemann zeta function and the unsolved Riemann hypothesis about the zeros of this function are covered in Chapter 20. Chapter 21 contains miscellaneous topics that are of mathematical interest.

Chapter 0 contains notational conventions used in this book. An impressive number of data sets available in Mathematica is also described. Chapter 1 provides an introduction to the fundamental two-dimensional plotting functions of Mathematica. The chapter also includes an introduction to the generation of animations. Chapter 2 uses elementary number theory to introduce a variety of basic and advanced features of Mathematica. Several important techniques are introduced, as well as different approaches to graphing data. Chapter 3 presents an introduction to the two-dimensional graphic capabilities of Mathematica. As an example generating various images related to rolling wheels is focused. The final section presents several unusual applications of the wheel concept such as the generation of a curve on which a square can roll smoothly and a construction of a drill that makes perfect square holes. Chapter 4 introduces three-dimensional graphics. The basic tools are contour plots, density plots and three-dimensional surface plots.

The dynamic display features of Mathematica are covered in Chapter 5. The center piece of this chapter is the “Manipulate” command. The Cantor set is explored in Chapter 6 with extensions to the complex world. Chapter 7 shows how the in-built functions of Mathematica can be used to investigate the result of iterating functions. Chapter 8 shows how to simulate a turtle and use it to generate approximations of a space filling curve.

Mathematica has very powerful visualization tools which allow one to generate surfaces defined by two parameters. Chapter 9 contains several applications: exploring some properties of torus, generating a double torus and looking at some unusual surfaces such as the Costa surface. The chapter concludes with a surprising extension, due to Mandelbrot, of the Koch snowflake construction to three dimensions.

Tiling of the plane has both decorative and mathematical uses and has been studied for centuries. Chapter 10 gives a study of this. In Chapter 11 the author investigates fractal sets. Chapter 12 discusses the solving of equations. Mathematica has a variety of functions to be used in optimization problems. Chapter 13 contains several case studies on this topic. Differential equations are dealt with in Chapter 14. Chapter 15 shows how a published construction in computational geometry – the construction of a three-dimensional room that contains a point invisible to guards placed at every vertex – must be changed if it is to be correct. Chapter 16 explains how some built-in functions can be used to do group theory. Chapter 17 contains a discussion of the four-colour theorem that seeks to turn Kempe’s false proof of 1879 into a viable algorithm for four-colouring planar graphs.

The main theme of Chapter 18 is the calculation of \(\pi\) using new formulas. The Banach-Tarski paradox is discussed in Chapter 19. The Riemann zeta function and the unsolved Riemann hypothesis about the zeros of this function are covered in Chapter 20. Chapter 21 contains miscellaneous topics that are of mathematical interest.

Reviewer: T. C. Mohan (Dehra Dun)

##### MSC:

65-01 | Introductory exposition (textbooks, tutorial papers, etc.) pertaining to numerical analysis |

11Yxx | Computational number theory |

68N15 | Theory of programming languages |

68W30 | Symbolic computation and algebraic computation |

65D18 | Numerical aspects of computer graphics, image analysis, and computational geometry |

11M06 | \(\zeta (s)\) and \(L(s, \chi)\) |