Mathematical Excursions
3rd Edition, June 2025
By David Salomon. A free eBook. xxxii+655 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
1.1 Curves and Surfaces 20
1.2 Perspective 38
1.3 Moir´e Patterns 38
1.4 Ruled Surfaces 40
1.5 Most Important Curve 40
1.6 Listings of Mathematica codes 41
2 Numbers: The Basic Building Blocks
2.1 Arithmetic Operations 47
2.2 Logical Operations 49
2.3 Integers 50
2.4 Rationals and Irrationals 77
2.5 Continued Fractions 84
2.6 Real Numbers 96
2.7 Complex Numbers 102
2.8 Hypercomplex Numbers? 110
2.9 Transcendental Numbers 112
2.10 Important and Interesting Numbers 113
2.11 Complex Golden Ratios 139
2.12 Approximating Formulas 146
2.13 Cyclic Numbers and Metadromes 147
3 Symmetry 151
3.1 A bit of History 153
3.2 Symmetry Groups 154
3.3 Orbifold Notation 173
3.4 The Magic Theorem 180
3.5 Orbifold Examples 181
3.6 Two-Dimensional Transformations 182
3.7 Symmetry in Tiling 206
3.8 Tessellations 212
3.9 Circle Inversions 214
3.10 Symmetry in text, speech, and ... 216
4 Infinity 223
4.1 A Short History of Infinity 224
4.2 Mathematical Infinity 225
4.3 Potential and Completed Infinities 227
4.4 Unexpected Results of Infinity 231
4.5 Set Theory 235
4.6 Physical Infinity 242
4.7 Infinitesimals and the Calculus 244
4.8 Restrictions in Mathematics 250
4.9 Debates in Mathematics 255
5 Order: Sequences and Series 259
5.1 Equations 260
5.2 The Pythagorean Theorem 261
5.3 A Different Dirac Equation 263
5.4 Sequences 264
5.5 Numerical Sequences 265
5.6 The Fibonacci Sequence 269
5.7 Metallic Ratios 284
5.8 The Comma Sequence 285
5.9 Quasi-Numeric Sequences 287
5.10 Series 288
5.11 The Real Harmonic Series 290
5.12 The Book-Stacking Problem 293
6 Paradoxes 29
6.1 Types of Paradoxes 299
6.2 Examples of Paradoxes 301
7 Probabilities: the Rule of Chance 325
7.1 Basic Concepts 325
7.2 More Probability Concepts 329
7.3 Randomness 331
7.4 Benford’s Law 339
7.5 Randomness in Dice 355
7.6 Go-First Dice 358
7.7 Subjective Probability 361
7.8 Probability and Psychology 362
7.9 The Birthday Paradox 364
7.10 Choosing a Candidate 366
7.11 Examples of Unexpected Probabilities 368
7.12 Probabilistic Counting and HLL 374
8 Geometry 381
8.1 Fractals 382
8.2 Weierstrass Function 399
8.3 Continuity 401
8.4 Interpolation 406
8.5 Least Squares Interpolation 407
8.6 Perlin Noise 412
8.7 Points and vectors 422
8.8 Representing Curves 425
8.9 PC Curves 427
8.10 Polynomial Interpolation 431
8.11 Spline Interpolation 435
8.12 Hermite Interpolation 436
8.13 Interactive Control 437
8.14 The Hermite Curve Segment 438
8.15 The Cubic Spline Curve 443
8.16 Cardinal Splines 448
8.17 Parabolic Blending: Catmull-Rom Curves 452
8.18 B´ezier Approximation 456
8.19 The B´ezier Curve 456
8.20 The Bernstein Form of the B´ezier Curve 459
8.21 Linear Perspective 463
8.22 Perspective: Basic Concepts 483
8.23 The Mathematics of Perspective 490
8.24 Slanted Squares with Integer Corners 500
8.25 Area of regular polygons 501
8.26 The Fourth Side of a Triangle? 502
9 Puzzles 503
9.1 Examples of Puzzles 503
10 Miscellaneous topics 531
10.1 The Gamma Function 531
10.2 Magic Squares 533
10.3 Parking as a greedy problem 539
10.4 Error-Control Codes 541
10.5 Compact Disc (CD) 545
10.6 Reed–Solomon Codes 547
10.7 SVD Image Compression 553
10.8 What is Average? 556
10.9 The power of the XOR 557
10.10 Brouwer fixed-point theorem 561
10.11 Short Topics 562
Bibliography 569
Answers to Exercises 579
Index 63
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