## MathematicaŽ in Action: Problem Solving Through Visualization and ComputationThis book is an example-based introduction to techniques, from elementary to advanced, of using Mathematica, a revolutionary tool for mathematical computation and exploration. By integrating the basic functions of mathematics with a powerful and easy-to-use programming language, Mathematica allows us to carry out projects that would be extremely laborious in traditional programming environments. And the new developments that began with version 6 — allowing the user to dyna- cally manipulate output using sliders or other controls — add amazing power to the interface. Animations have always been part of Mathematica, but the new design allows the manipulation of any number of variables, an important enhancement. Mathematica in Action illustrates this power by using demonstrations and ani- tions, three-dimensional graphics, high-precision number theory computations, and sophisticated geometric and symbolic programming to attack a diverse collection of problems.. It is my hope that this book will serve a mathematical purpose as well, and I have interspersed several unusual or complicated examples among others that will be more familiar. Thus the reader may have to deal simultaneously with new mat- matics and new Mathematica techniques. Rarely is more than undergraduate mathematics required, however. An underlying theme of the book is that a computational way of looking at a mathematical problem or result yields many benefits. For example: Well-chosen computations can shed light on familiar relations and reveal new patterns. One is forced to think very precisely; gaps in understanding must be eliminated if a program is to work. |

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### Contents

23 | |

2 Prime Numbers | 53 |

3 Rolling Circles | 77 |

4 ThreeDimensional Graphs | 113 |

5 Dynamic Manipulations | 141 |

6 The Cantor Set Real and Complex | 168 |

7 The Quadratic Map | 179 |

8 The Recursive Turtle | 209 |

14 Differential Equations | 363 |

15 Computational Geometry | 399 |

16 Check Digits and the Pentagon | 423 |

17 Coloring Planar Maps | 430 |

18 New Directions For | 473 |

19 The BanachTarski Paradox | 491 |

20 The Riemann Zeta Function | 505 |

21 Miscellany | 523 |

9 Parametric Plotting of Surfaces | 235 |

10 Penrose Tiles | 267 |

Julia Sets and the Mandelbrot Set | 276 |

12 Solving Equations | 301 |

13 Optimization | 329 |

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### Common terms and phrases

_ 2 _ algorithm approximation ArcTan AspectRatio Axes False Benford's Law Black Boxed False Cantor set chapter circle color ColorFunction complex numbers computation ContourPlot convergence CountryData curve cycloid default define digits Dynamic EdgeForm edges equation Evaluate example FaceForm Fibonacci FindMinimum FindRoot Flatten formula Frame True FrameTicks Automatic FrameTicks Range function Gaussian primes Graphics integer interval iteration Julia set Kempe chain Line ListLinePlot Mandelbrot set Manipulate Mathematica MaxRecursion Mesh MeshFunctions method NDSolve Nest NestList option orbit output package parameter ParametricPlot planar graphs plane plot Plot3D PlotLabel PlotPoints PlotRange PlotStyle Thick points PointSize poly Polygon precision PrimeProduct PrimeQ problem RegionFunction RGBColor Riemann hypothesis sequence shows soln solution Solve space-filling curve square starting value Table theorem tiling ToRules torus trajectory Transpose triangle vector vertex vertices ViewPoint Villarceau circles zeros