By David Salomon. A free eBook. xxxiv+716 pages, available here.
An errata list and other information may later be added to this website.
About This Book
This book 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.
Table of Contents
Preface ix
Introduction 1
1 Graphics: Visible Math Objects 19
1.1 Curves and Surfaces 20
1.2 Perspective 38
1.3 Moire Patterns 39
1.4 Ruled Surfaces 41
1.5 Most Important Curve 41
1.6 Listings of Mathematica codes 42
2 Numbers: The Basic Building Blocks 47
2.1 Arithmetic Operations 47
2.2 Logical Operations 49
2.3 Integers 50
2.4 Rationals and Irrationals 78
2.5 Continued Fractions 85
2.6 Real Numbers 98
2.7 Complex Numbers 103
2.8 Hypercomplex Numbers? 112
2.9 Quaternions 115
2.10 Transcendental Numbers 129
2.11 Important and Interesting Numbers 130
2.12 Complex Golden Ratios 156
2.13 Approximating Formulas 163
2.14 Cyclic Numbers and Metadromes 164
3 Symmetry 167
3.1 A bit of History 169
3.2 Symmetry Groups 170
3.3 Orbifold Notation 189
3.4 The Magic Theorem 196
3.5 Orbifold Examples 197
3.6 Two-Dimensional Transformations 198
3.7 Symmetry in Tiling 222
3.8 Tessellations 228
3.9 Circle Inversions 230
3.10 Symmetry in text, speech, and ... 232
4 Infinity 239
4.1 A Short History of Infinity 240
4.2 Mathematical Infinity 241
4.3 Potential and Completed Infinities 243
4.4 Unexpected Results of Infinity 247
4.5 Set Theory 251
4.6 Physical Infinity 259
4.7 Infinitesimals and the Calculus 261
4.8 Restrictions in Mathematics 267
4.9 Debates in Mathematics 272
5 Order: Sequences and Series 275
5.1 Equations 276
5.2 The Pythagorean Theorem 277
5.3 A Different Dirac Equation 279
5.4 Sequences 280
5.5 Numerical Sequences 281
5.6 The Fibonacci Sequence 285
5.7 Metallic Ratios 300
5.8 The Comma Sequence 301
5.9 Quasi-Numeric Sequences 303
5.10 Series 304
5.11 The Real Harmonic Series 309
5.12 The Book-Stacking Problem 312
6 Paradoxes 317
6.1 Types of Paradoxes 317
6.2 Examples of Paradoxes 319
7 Probabilities: the Rule of Chance 345
7.1 Basic Concepts 345
7.2 More Probability Concepts 349
7.3 Randomness 352
7.4 Benford’s Law 361
7.5 Randomness in Dice 376
7.6 Go-First Dice 380
7.7 Subjective Probability 382
7.8 Probability and Psychology 383
7.9 The Birthday Paradox 385
7.10 Choosing a Candidate 387
7.11 Examples of Unexpected Probabilities 389
7.12 Probabilistic Counting and HLL 402
8 Geometry 409
8.1 Fractals 410
8.2 Weierstrass Function 427
8.3 Continuity 429
8.4 Interpolation 434
8.5 Least Squares Interpolation 435
8.6 Perlin Noise 440
8.7 Three-Dimensional Rotations 450
8.8 Vector Operations 451
8.9 Points and vectors 469
8.10 Representing Curves 472
8.11 PC Curves 474
8.12 Polynomial Interpolation 477
8.13 Spline Interpolation 481
8.14 Hermite Interpolation 483
8.15 Interactive Control 484
8.16 The Hermite Curve Segment 485
8.17 The Cubic Spline Curve 490
8.18 Cardinal Splines 495
8.19 Parabolic Blending: Catmull-Rom Curves 499
8.20 Bezier Approximation 503
8.21 The Bezier Curve 503
8.22 The Bernstein Form of the Bezier Curve 506
8.23 Linear Perspective 510
8.24 Perspective: Basic Concepts 530
8.25 The Mathematics of Perspective 537
8.26 Slanted Squares with Integer Corners 547
8.27 Area of regular polygons 548
8.28 The Fourth Side of a Triangle? 549
9 Puzzles 551
9.1 Examples of Puzzles 551
10 Miscellaneous topics 581
10.1 The Gamma Function 581
10.2 Magic Squares 583
10.3 Parking as a greedy problem 589
10.4 Error-Control Codes 591
10.5 Compact Disc (CD) 596
10.6 Reed–Solomon Codes 598
10.7 SVD Image Compression 604
10.8 What is Average? 607
10.9 The power of the XOR 608
10.10 Brouwer fixed-point theorem 611
10.11 Short Topics 613
Bibliography 621
Answers to Exercises 633
Index 692
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