Mathematica® in Action

0 A Brief Introduction.- 0.1 Notational Conventions.- 0.2 Typesetting.- 0.3 Basic Mathematical Functions.- 0.4 Using Functions.- 0.5 Replacements.- 0.6 Lists.- 0.7 Getting Information.- 0.8 Algebraic Manipulations.- 0.9 Customizing Mathematica.- I Basic Concepts.- 1 Plotting.- 1.1 Plot.- 1.2 An ArcSin Curiosity.- 1.3 Adapting and Suppressing.- 1.4 Plotting Tables and Tabling Plots.- 1.5 Dealing with Discontinuities.- 1.6 ListPlot.- 1.7 ParametricPlot.- 2 Prime Numbers.- 2.1 Basic Number Theory Functions.- 2.2 Where the Primes Are.- 2.3 The Prime Number Race.- 2.4 Euclid and Fibonacci.- 2.5 Strong Pseudoprimes.- 3 Rolling Circles.- 3.1 Discovering the Cycloid.- 3.2 The Derivative of the Trochoid.- 3.3 Abe Lincoln's Somersaults.- 3.4 The Cycloid's Intimate Relationship with Gravity.- 3.5 Bicycles, Square Wheels, and Square Drills.- 4 Surfaces.- 4.1 Using Two-Dimensional Tools.- 4.2 Plotting Surfaces.- 4.3 Mixed Partial Derivatives Need Not Be Equal.- 4.4 Failure of the Only-Critical-Point-in-Town Test.- 4.5 Raising Contours to New Heights.- II Graphics Issues and Applications.- 5 The Cantor Set, Real and Complex.- 5.1 The Real Cantor Set.- 5.2 The Cantor Function.- 5.3 Complex Cantor Sets.- 6 The Quadratic Map.- 6.1 Iterating Functions.- 6.2 Numerical Subtleties.- 6.3 Chaos and the Quadratic Map.- 6.4 Bifurcations.- 7 The Recursive Turtle.- 7.1 The Literate Turtle.- 7.2 Space-Filling Curves.- 7.3 A Surprising Application.- 7.4 Trees, Mathematical and Botanical.- 8 Parametric Plotting of Surfaces.- 8.1 Introduction to ParametricPlot3D.- 8.2 Dandelin Spheres.- 8.3 Optimizing a Root Cellar.- 8.4 Adaptive Surface Plotting.- 8.5 A Fractal Tetrahedron.- 9 Penrose Tiles.- 9.1 Nonperiodic Tilings.- 9.2 Penrose Tilings.- 9.3 Penrose Rhombs.- 10 Fractals, Ferns, and Julia Sets.- 10.1 Iterated Function Systems.- 10.2 Biasing the Chaos Game: Barnsley's Fern.- 10.3 Julia Sets.- III Numerical Mathematics.- 11 Custom Curves.- 11.1 Basic Curve Fitting.- 11.2 Applications.- 11.3 Bézier Curves.- 11.4 A Virtual Bicycle Ride.- 11.5 Two-Dimensional Annotation of a Three-Dimensional Image.- 11.6 Stineman Interpolation.- 12 Solving Equations.- 12.1 Solve.- 12.2 LinearSolve.- 12.3 NSolve.- 12.4 FindRoot.- 12.5 Radicals and Beyond.- 12.6 FindAllCrossings.- 12.7 FindAllCrossings2D.- 13 Differential Equations.- 13.1 Solving Differential Equations.- 13.2 Stylish Plots.- 13.3 Pitfalls of Numerical Computing.- 13.4 Basins of Attraction.- IV Number Theory.- 14 Public-Key Encryption.- 14.1 Random Primes.- 14.2 The RSA Scheme.- 14.3 Character Analysis.- 15 Egyptian Fractions.- 15.1 A Greedy Algorithm.- 15.2 A Puzzling Halting Problem.- 15.3 Splitting Fractions.- 16 The Ancient and Modern Euclidean Algorithm.- 16.1 The Oldest Surviving Algorithm.- 16.2 The Extended Euclidean Algorithm.- 16.3 Continuants.- 16.4 Euclid and Diophantus.- 16.5 The Chinese Remainder Theorem.- 16.6 Rationals.- 16.7 Continued Fractions.- 17 Imaginary Primes and Prime Imaginaries.- 17.1 The Gaussian Integers.- 17.2 Residues and Nonresidues.- 17.3 A Surprising Application of Imaginary Primes.- 17.4 The Gaussian Primes.- 17.5 Sums of Two Squares.- 17.6 Primes as Stepping-Stones.- 18 Certifying Primality.- 18.1 P and NP.- 18.2 A New Look at an Old Primality Test.- 18.3 Finding Pratt Certificates.- 18.4 Perrin's Pretty Primality Test.- 18.5 The Lucas Primality Test.- 19 Check Digits and the Pentagon.- 19.1 The Group of the Pentagon.- 19.2 The Perfect Dihedral Method.- V Advanced Projects.- 20 New Directions for ?.- 20.1 The Classical Theory of ?.- 20.2 The Postmodern Theory of ?.- 20.3 A Most Depressing Proof.- 20.4 Variations on the Theme.- 20.5 A Strange Coincidence?.- 21 Rearrangement of Series.- 21.1 Harmonie Preliminaries.- 21.2 The Alternating Harmonic Series.- 22 Escher's Patterns.- 22.1 Escher's Motif.- 22.2 Escher Counts.- 22.3 Programming the Patterns.- 22.4 Subtle Patterns.- 22.5 The Coloring Equations.- 22.6 More Motifs.- 23 Computational Geometry.- 23.1 Basic Computational Geometry.- 23.2 The Art Gallery Theorem.- 23.3 A Very Strange Room.- 23.4 More Euclid.- 24 Coloring Planar Maps and Graphs.- 24.1 Preliminaries: Combinatorica.- 24.2 Planar Maps.- 24.3 Euler's Formula.- 24.4 Displaying Graphs.- 24.5 Kempe's Attempt.- 24.6 The Implementation.- 24.7 Random Ideas.- 25 The Riemann Zeta Function.- 25.1 The Riemann Zeta Function.- 25.2 The Influence of the Zeros of ? on the Distribution of Primes.- 26 The Banach-Tarski Paradox.- 26.1 The Paradoxical Free Product ? 2*? 3.- 26.2 A Hyperbolic Representation of the Group.- 26.3 The Geometrical Paradox.- References.