Mathematica in Action: Problem Solving Through Visualization and Computation

591 Pages·2010·20.16 MB·English

by  Stan Wagon (auth.)

Mathematica in Action

Stan Wagon

Mathematica in Action

Problem Solving Through Visualization
and Computation

Third edition
Stan Wagon
Department of Mathematics
and Computer Science
Macalester College
1600 Grand Avenue
St. Paul, MN 55105
USA
[email protected]
Wolfram Mathematica ® is a registered trademark of Wolfram Research, Inc.

ISBN 978-0-387-75366-9 e-ISBN 978-0-387-75477-2
DOI 10.1007/978-0-387-75477-2
Springer New York Dordrecht Heidelberg London

Library of Congress Control Number: 2010928640

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Contents
Preface ix
0 A Brief Introduction 1
0.1 Notational Conventions 2
0.2 Typesetting 3
0.3 Basic Mathematical Functions 5
0.4 Using Functions 7
0.5 Replacements 12
0.6 Lists 13
0.7 Getting Information 15
0.8 Algebraic Manipulations 17
0.9 Customizing Mathematica 19
0.10 Comprehensive Data Sets in Mathematica 19
1 Plotting 23
1.1 Plot 24
1.2 An Arcsin Curiosity 26
1.3 Adaptive Plotting 28
1.4 Plotting Tables and Tabling Plots 31
1.5 Dealing with Discontinuities 34
1.6 ListPlot 37
1.7 ParametricPlot 42
1.8 Difficult Plots 48
2 Prime Numbers 53
2.1 Basic Number Theory Functions 54
2.2 Where the Primes Are 60
2.3 The Prime Number Race 66
2.4 Euclid and Fibonacci 70
2.5 Strong Pseudoprimes 73
3 Rolling Circles 77
3.1 Discovering the Cycloid 78
3.2 The Derivative of the Trochoid 82
3.3 Abe Lincoln’s Somersaults 84
3.4 The Cycloid’s Intimate Relationship with Gravity 90
3.5 Bicycles, Square Wheels, and Square-Hole Drills 98
vi Contents
4 Three-Dimensional Graphs 113
4.1 Using Two-Dimensional Tools 114
4.2 Plotting Surfaces 119
4.3 Mixed Partial Derivatives Need Not Be Equal 130
4.4 Failure of the Only-Critical-Point-in-Town Test 134
4.5 Raising Contours to New Heights 137
4.6 A New View of Pascal’s Triangle 139
5 Dynamic Manipulations 141
5.1 Basic Manipulations 142
5.2 Control Variations 144
5.3 Locators 148
5.4 Fine Control 151
5.5 Three Case Studies 157
6 The Cantor Set, Real and Complex 169
6.1 The Real Cantor Set 170
6.2 The Cantor Function 173
6.3 Complex Cantor Sets 175
7 The Quadratic Map 179
7.1 Iterating Functions 180
7.2 The Four Flavors of Real Numbers 187
7.3 Attracting and Repelling Cycles 192
7.4 Measuring Instability: The Lyapunov Exponent 199
7.5 Bifurcations 202
8 The Recursive Turtle 209
8.1 The Literate Turtle 210
8.2 Space-Filling Curves 215
8.3 A Surprising Application 223
8.4 Trees, Mathematical and Botanical 233
9 Parametric Plotting of Surfaces 235
9.1 Introduction to ParametricPlot3D 236
9.2 A Classic Torus Dissection 243
9.3 The Villarceau Circles 250
9.4 Beautiful Surfaces 254
9.5 A Fractal Tetrahedron 261
Contents vii
10 Penrose Tiles 267
10.1 Nonperiodic Tilings 268
10.2 Penrose Tilings 270
10.3 Penrose Rhombs 274
11 Complex Dynamics: Julia Sets and the
Mandelbrot set (by Mark McClure) 277
11.1 Complex Dynamics 278
11.2 Julia Sets and Inverse Iteration 284
11.3 Escape Time Algorithms and the Mandelbrot Set 292
12 Solving Equations 301
12.1 Solve 302
12.2 Diophantine Equations 306
12.3 LinearSolve 310
12.4 NSolve 312
12.5 FindRoot 314
12.6 Finding All Roots in an Interval 315
12.7 FindRoots2D 318
12.8 Two Applications 322
13 Optimization 329
13.1 FindMinimum 330
13.2 Algebraic Optimization 333
13.3 Linear Programming and Its Cousins 334
13.4 Case Study: Interval Methods for a SIAM Challenge 346
13.5 Case Study: Shadowing Chaotic Maps 353
13.6 Case Study: Finding the Best Cubic 360
14 Differential Equations 363
14.1 Solving Differential Equations 364
14.2 Stylish Plots 367
14.3 Pitfalls of Numerical Computing 376
14.4 Basins of Attraction 382
14.5 Modeling 385
15 Computational Geometry 399
15.1 Basic Computational Geometry 400
15.2 The Art Gallery Theorem 404
15.3 A Very Strange Room 406
15.4 More Euclid 413
viii Contents
16 Check Digits and the Pentagon 423
16.1 The Group of the Pentagon 424
16.2 The Perfect Dihedral Method 427
17 Coloring Planar Maps 431
17.1 Introduction to Combinatorica 432
17.2 Planar Maps 437
17.3 Euler’s Formula 441
17.4 Kempe’s Attempt 445
17.5 Kempe Resurrected 449
17.6 Map Coloring 458
17.7 A Great Circle Conjecture 468
18 New Directions for Π 473
18.1 The Classical Theory of Π 474
18.2 The Postmodern Theory of Π 480
18.3 A Most Depressing Proof 483
18.4 Variations on the Theme 488
19 The Banach-Tarski Paradox 491
19.1 A Paradoxical Free Product 492
19.2 A Hyperbolic Representation of the Group 495
19.3 The Geometrical Paradox 499
20 The Riemann Zeta Function 505
20.1 The Riemann Zeta Function 506
20.2 The Influence of the Zeros of Ζ on the Distribution of Primes 512
20.3 A Backwards Look at Riemann’s R x 519
21 Miscellany 523
(cid:72) (cid:76)
21.1 An Educational Integral 524
21.2 Making the Alternating Harmonic Series Disappear 525
21.3 Bulletproof Prime Numbers 528
21.4 Gaussian Moats 530
21.5 Frobenius Number by Graphs 536
21.6 Benford’s Law of First Digits 542
References 557
Mathematica Index 566
Subject Index 572
Preface
This 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 dynami-
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 anima-
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 mathe-
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:
(cid:159) Well-chosen computations can shed light on familiar relations and reveal new
patterns.
(cid:159) One is forced to think very precisely; gaps in understanding must be eliminated if
a program is to work.
(cid:159) Dozens (or hundreds or thousands) of cases can be examined, perhaps showing
new patterns or phenomena.
(cid:159) Methods of verifying the results must be worked out, again adding to one's
overall understanding.
(cid:159) Different proofs of the same theorem can be compared from the point of view of
algorithmic efficiency.