By David Salomon. A free eBook. xxx+574 pages, available here.
An errata list and other information may later be added to this website.
About This Book
This book was started as a collection of beautiful facts, objects, theorems, and relations in mathematics. Over time, however, as more and more material was added, it became simply a place to summarize, discuss, and explain mathematical topics that are of personal interest to me. As a result, the book is personal (some may say that it is a hodgepodge of topics and facts). An occasional reader may find certain topics that are of interest and may skip the rest of the book. In any case, it is free.
The text of this book includes many references. They have the format [name date] and are listed in detail in the bibliography at the end of the book. Any errors, mistakes, misprints, and mistypes found here, as well as any criticism and suggestions, can be emailed to me at [email protected].
This book is aimed toward (1) those who would like to give mathematics another try and, (2) those who locate (in the table of contents below) a subject that looks interesting. Code examples are mostly the Mathematica software, with some in GeoGebra and Desmos. The code snippets included in the book are meant for readability, not efficiency. They can serve as basic building blocks on which readers can build more complex programs.
Any comments, suggestions, and corrections are welcome and should be emailed to the author at [email protected]. However, if you notice something missing, consider the following quote (from Mark Twain) “A successful book is not made of what is in it, but of what is left out of it.”
Organization and Features
Chapter 1 tries to whet the appetite of the reader with art; it displays mathematical objects graphically.
Chapter 2 mentions types as well as many interesting properties of numbers.
Chapter 3 introduces symmetry. It presents many examples of symmetry and shows how to classify symmetries.
Chapter 4 is a discussion of the widely misunderstood concept of infinity, its 'mysteries,' pitfalls, and unexpected applications.
Chapter 5 introduces the reader to mathematical sequences and series, most notably, the ones associated with the name of Fibonacci.
Chapter 6 is concerned with the 'paradoxical' concept of paradoxes. The main classes of paradoxes are listed and examples are shown.
Chapter 7 covers the all-important and widely misunderstood topic of probability. Especially interesting is the discussion of randomness.
Chapter 8 deals with topics in geometry, but not the traditional ones of shapes, axioms, and proofs. The stress instead is on interpolation by means of splines.
The short Chapter 9 lists many brain teaser and funny puzzles.
Finally, Chapter 10 is a collection of miscellaneous topics, among them magic squares and error-control codes.
Preface ix
Introduction 1
1 Graphics: Visible Math Objects 17
1.1 Curves and Surfaces 18
1.2 Perspective 36
1.3 Ruled Surfaces 37
1.4 Most Important Curve 37
1.5 Listings of Mathematica codes 39
2 Numbers: The Basic Building Blocks 43
2.1 Arithmetic Operations 43
2.2 Logical Operations 45
2.3 Integers 46
2.4 Rationals and Irrationals 70
2.5 Real Numbers 80
2.6 Complex Numbers 84
2.7 Hypercomplex Numbers? 92
2.8 Transcendental Numbers 94
2.9 Important and Interesting Numbers 95
2.10 Complex Golden Ratios 118
2.11 Approximating Formulas 124
2.12 Cyclic Numbers and Metadromes 127
3 Symmetry 131
3.1 A bit of History 133
3.2 Symmetry Groups 134
3.3 Orbifold Notation 153
3.4 The Magic Theorem 160
3.5 Orbifold Examples 161
3.6 Two-Dimensional Transformations 162
3.7 Symmetry in Tiling 186
3.8 Tessellations 191
3.9 Circle Inversions 194
3.10 Symmetry in text, speech, and ... 196
4 Infinity 201
4.1 A Short History of Infinity 202
4.2 Mathematical Infinity 203
4.3 Potential and Completed Infinities 205
4.4 Unexpected Results of Infinity 209
4.5 Set Theory 212
4.6 Physical Infinity 219
4.7 Infinitesimals and the Calculus 221
5 Order: Sequences and Series 229
5.1 Equations 230
5.2 The Pythagorean Theorem 231
5.3 A Different Dirac Equation 233
5.4 Sequences 234
5.5 Numerical Sequences 235
5.6 The Fibonacci Sequence 238
5.7 Metallic Ratios 253
5.8 The Comma Sequence 254
5.9 Quasi-Numeric Sequences 256
5.10 Series 257
5.11 The Real Harmonic Series 259
5.12 The Book-Stacking Problem 262
6 Paradoxes 267
6.1 Types of Paradoxes 267
6.2 Examples of Paradoxes 268
7 Probabilities: the Rule of Chance 289
7.1 Basic Concepts 289
7.2 More Probability Concepts 293
7.3 Randomness 295
7.4 Benford’s Law 303
7.5 Randomness in Dice 305
7.6 Go-First Dice 308
7.7 Subjective Probability 310
7.8 Probability and Psychology 311
7.9 The Birthday Paradox 314
7.10 Choosing a Candidate 315
7.11 Examples of Unexpected Probabilities 317
7.12 Probabilistic Counting and HLL 323
8 Geometry 329
8.1 Fractals 330
8.2 Weierstrass Function 340
8.3 Continuity 343
8.4 Interpolation 347
8.5 Least Squares Interpolation 348
8.6 Perlin Noise 354
8.7 Points and vectors 364
8.8 Representing Curves 367
8.9 PC Curves 369
8.10 Polynomial Interpolation 372
8.11 Spline Interpolation 376
8.12 Hermite Interpolation 377
8.13 Interactive Control 378
8.14 The Hermite Curve Segment 379
8.15 The Cubic Spline Curve 385
8.16 Cardinal Splines 390
8.17 Parabolic Blending: Catmull-Rom Curves 393
8.18 Bezier Approximation 397
8.19 The Bezier Curve 398
8.20 The Bernstein Form of the B´ezier Curve 400
8.21 Linear Perspective 404
8.22 Perspective: Basic Concepts 424
8.23 The Mathematics of Perspective 430
8.24 Slanted Squares with Integer Corners 441
8.25 Area of regular polygons 442
8.26 The Fourth Side of a Triangle? 443
9 Puzzles 445
9.1 Examples of Puzzles 445
10 Miscellaneous topics 463
10.1 The Gamma Function 463
10.2 Magic Squares 465
10.3 Parking as a greedy problem 471
10.4 Error-Control Codes 473
10.5 Compact Disc (CD) 477
10.6 Reed–Solomon Codes 479
10.7 What is Average? 485
10.8 The power of the XOR 487
10.9 Brouwer fixed-point theorem 490
10.10 Short Topics 492
Bibliography 497
Answers to Exercises 507
Index 553
The content of most textbooks is perishable, but the tools of self-directness serve one well over time. —Albert Bandura.
What I have to say about this book can be found inside the book. --Albert Einstein