By David Salomon. A free eBook. xxxiv+912 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.
The following AI persona have contributed images, poems, some Mathematica code, and much other1 help to this edition. Their help is always explicitly acknowledged in the text.
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.
Chapter 10 is a survey of several important error-correction methods. Of special interest is the details of error-correction in the popular QR codes.
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 A Variable Mesh 41
1.6 Most Important Curve 43
1.7 Listings of Mathematica codes 43
2 Numbers: The Basic Building Blocks 49
2.1 Arithmetic Operations 49
2.2 Logical Operations 51
2.3 Integers 52
2.4 Rationals and Irrationals 83
2.5 Continued Fractions 90
2.6 Rational Approximations 103
2.7 Real Numbers 107
2.8 Complex Numbers 112
2.9 Hypercomplex Numbers? 121
2.10 Quaternions 124
2.11 Transcendental Numbers 138
2.12 Important and Interesting Numbers 140
2.13 Complex Golden Ratios 168
2.14 Approximating Formulas 175
2.15 Cyclic Numbers and Metadromes 176
3 Symmetry 179
3.1 A bit of History 181
3.2 Symmetry Groups 182
3.3 Orbifold Notation 201
3.4 The Magic Theorem 208
3.5 Orbifold Examples 210
3.6 Two-Dimensional Transformations 210
3.7 Symmetry in Tiling 234
3.8 Tessellations 240
3.9 Circle Inversions 243
3.10 Symmetry in text, speech, and ... 244
4 Infinity 251
4.1 A Short History of Infinity 252
4.2 Mathematical Infinity 253
4.3 Potential and Completed Infinities 255
4.4 Unexpected Results of Infinity 259
4.5 Set Theory 263
4.6 Physical Infinity 271
4.7 Infinitesimals and the Calculus 273
4.8 Restrictions in Mathematics 279
4.9 Debates in Mathematics 285
5 Order: Sequences and Series 289
5.1 Equations 290
5.2 The Pythagorean Theorem 291
5.3 A Different Dirac Equation 293
5.4 Sequences 294
5.5 Numerical Sequences 295
5.6 The Fibonacci Sequence 299
5.7 Metallic Ratios 318
5.8 The Inverse Palindromic Series 319
5.9 The Comma Sequence 320
5.10 Quasi-Numeric Sequences 322
5.11 Series 325
5.12 The Real Harmonic Series 331
5.13 The Book-Stacking Problem 334
6 Paradoxes 339
6.1 Types of Paradoxes 339
6.2 Examples of Paradoxes 341
7 Probabilities: the Rule of Chance 367
7.1 Basic Concepts 367
7.2 More Probability Concepts 371
7.3 Randomness 374
7.4 Benford’s Law 384
7.5 Randomness in Dice 399
7.6 Go-First Dice 403
7.7 Subjective Probability 405
7.8 Probability and Psychology 406
7.9 The Birthday Paradox 408
7.10 The Drunken Man 410
7.11 Choosing a Candidate 411
7.12 Examples of Unexpected Probabilities 413
7.13 Probabilistic Counting and HLL 429
8 Geometry 435
8.1 Fractals 436
8.2 Weierstrass Function 453
8.3 Continuity 455
8.4 Interpolation 460
8.5 Least Squares Interpolation 461
8.6 Perlin Noise 466
8.7 Three-Dimensional Rotations 476
8.8 Vector Operations 477
8.9 Points and vectors 495
8.10 Representing Curves 498
8.11 PC Curves 500
8.12 Polynomial Interpolation 503
8.13 Spline Interpolation 507
8.14 Hermite Interpolation 509
8.15 Interactive Control 510
8.16 The Hermite Curve Segment 511
8.17 The Cubic Spline Curve 516
8.18 Cardinal Splines 521
8.19 Parabolic Blending: Catmull–Rom Curves 525
8.20 Bezier Approximation 529
8.21 The Bezier Curve 529
8.22 The Bernstein Form of the Bezier Curve 532
8.23 Linear Perspective 536
8.24 Perspective: Basic Concepts 556
8.25 The Mathematics of Perspective 563
8.26 Slanted Squares with Integer Corners 573
8.27 Area of regular polygons 574
8.28 The Fourth Side of a Triangle? 575
9 Puzzles 577
9.1 Examples of Puzzles 577
10 Error-Control Codes 607
10.1 Error-Control Codes 607
10.2 Galois (Finite) Fields and Their Algebra 613
10.3 Hamming codes 645
10.4 Compact Disc (CD) 655
10.5 Reed–Solomon Codes 657
10.6 Error Correction in QR Codes 692
10.7 Reed–Muller Codes 716
10.8 Golay Codes 740
10.9 OLS Error-Correction 756
11 Miscellaneous Topics 769
11.1 The Gamma Function 769
11.2 Magic Squares 771
11.3 Parking as a greedy problem 777
11.4 SVD Image Compression 779
11.5 What is Average? 782
11.6 The power of the XOR 783
11.7 Brouwer fixed-point theorem 787
11.8 Rational Points on a Circle 790
11.9 Short Topics 792
Bibliography 799
Answers to Exercises 813
Index 884
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