The Set of Math-sterpieces

Introduction

Mathematics, rightly viewed, possesses not only truth, but supreme beauty cold and austere, like that of sculpture, without appeal to any part of our weaker nature, without the gorgeous trappings of painting or music, yet sublimely pure, and capable of a stern perfection such as only the greatest art can show. The true spirit of delight, the exaltation, the sense of being more than Man, which is the touchstone of the highest excellence, is to be found in mathematics as surely as in poetry.
— Bertrand Russell

As said by Russell, mathematics possesses this beauty and creativity, and that has made it my favourite hobby. While there are many ways to learn math, particularly through formalism and abstraction, both of which are essential for any serious mathematical endeavours, I find a special, almost unexplainable comfort in reading pop-maths books and their fascinating stories. And, when those books are done right, they are perfect for getting curious people into math, especially the ones who want to dive in but don’t know where to start or feel intimidated.

The purpose of this blog is twofold: first, to serve as a curated wishlist of such resources, in the hope that it might inspire others to revisit mathematics. And second, you may have noticed the absence of explicit math blogs here, despite having math-y references scattered throughout my other non-mathematical posts. This is intentional, as there are already so many amazing resources out there, I didn’t feel the need to just repeat what’s already been done.

Legendary Books

Must-Reads

A Mathematician’s Apology (G. H. Hardy)

A short essay where Hardy (mentor of Srinivasa Ramanujan) discusses the beauty and usefulness of pure mathematics.
I wanted to start with this recommendation because the books mentioned may not directly help with exams or jobs, but they are meant more to explore the beauty of mathematics. I hope this inspires an appreciation for math, and I apologize if it’s not what you were looking for.

Oliver Byrne’s Elements of Euclid

A beautiful work of art, where Byrne reimagined the first six books of The Elements of Euclid, replacing letters with vividly coloured diagrams and symbols. [Website/PDF]

Proofs from THE BOOK (Martin Aigner, Günter M. Ziegler)

The famous math nomad, Paul Erdős, once said that you need not believe in God but, as a mathematician, you should believe in the existence of THE BOOK, which contains the most elegant and perfect proofs of all mathematical theorems. Is it this THE BOOK?

Gödel, Escher, Bach: an Eternal Golden Braid (Douglas R. Hofstadter)

Gödel, a mathematician; Escher, a graphic artist; and Bach, a music composer. What is the connection here? Dive in and find out!

Treatises

Principia Mathematica (Alfred North Whitehead, Bertrand Russell)

This three-volume series is arguably the most important work in the history of mathematics and philosophy. And it also contains the infamous 360 page proof of \(1+1=2\).

Contributions to the Founding of the Theory of Transfinite Numbers (Georg Cantor)

Contains important developments related to transfinite numbers, which are nothing but infinite numbers

The Mathematical Analysis of Logic (George Boole)

The introduction of Calculus of Deductive Reasoning, aka Boolean Algebra

A Treatise on Probability (John Maynard Keynes)

A “logical-relationist” approach to the theory of uncertainty, which has started to gain more traction due to advancements in artificial intelligence

The Road to Reality: A Complete Guide to the Laws of the Universe (Roger Penrose)

The most ambitious work of science, providing a comprehensive account of the Standard Model of particle physics while also developing the mathematical theory required from scratch

Handbooks

A Synopsis of Elementary Results in Pure and Applied Mathematics (George Shoobridge Carr)

The holy two-volume set that Ramanujan used to teach himself math.

The World of Mathematics (James R. Newman)

The monumental four-volume set which includes a general survey of mathematics, historical and biographical information on prominent mathematicians throughout history

The Princeton Companion to Mathematics (Timothy Gowers, June Barrow-Green, Imre Leader)

A one-of-a-kind reference for anyone with a serious interest in mathematics, covering popular mathematicians, concepts, theorems and most importantly expository introductions to many branches of mathematics.

Fiction

My taste in fictional math books is the same as A.A.: only the very best, with just the right amount of math

Flatland: A Romance of Many Dimensions (Edwin A. Abbott)

A pioneering work of mathematical fiction, exploring the concept of dimensions through the eyes of our protagonist, Square, who is a Flatlander living in a 2-D world, Flatland

Surreal Numbers (Donald Knuth)

A conversation between a young couple turns into reinventing numbers

Gallery of the Infinite (Richard Evan Schwartz)

An illustrated comic, introducing important concepts of Set Theory in a playful manner

Lost in the Math Museum: A Survival Story (Colin Adams)

The imaginative journey of a 16-year-old girl trying to escape death in a mathematics museum, leading her through a series of mathematical theorems, paradoxes, and curiosities.

Miniatures

Miniatures are my favourite way to enjoy mathematics, they are a collection of short but entertaining tidbits from all across a particular field, which can be consumed at one’s convenience. With the exception of the first two fictional works and the last section, the majority of the forthcoming books fall under this category.

For Newbies

Collections

Math Wonders: To Inspire Teachers and Students (Alfred Posamentier)

The book that inspired me into learning mathematics. It is just a collection of fun patterns, calculations and other entertainments across high-school math topics.

The Math Book: Big Ideas Simply Explained (DK)

A visually rich encyclopedia which covers everything from ancient times all the way to modern-day math ideas.

Humble Pi (Matt Parker)

A highly entertaining read dissecting cases when maths goes wrong in the real world

The Proof and the Pudding (Jim Henle)

This one is for foodies, a witty and flavourful blend of mathematical treats and gastronomic delights (yes! with recipes) that reveal the parallels between the mathematical world and the art of cooking.

A Passion For Mathematics: Numbers, Puzzles, Madness, Religion, And The Quest For Reality (Clifford A. Pickover)

Super short math tidbits, suitable for all kinds of audience

How Round Is a Cube?: And Other Curious Mathematical Ponderings (James Tanton)

A set of 34 super approachable essays on all sorts of math topics, packed with great visuals, part of the Mathematical Circles Library which aims to foster a greater awareness and appreciation of mathematics among young students.

To save time, I won’t be adding my thoughts for every book, especially when their titles are kinda self-explanatory or they are similar to the ones already discussed. So, just dropping some titles below!

• Beautiful Geometry (Eli Maor, Eugen Jost)
• Mathematics Galore! (James Tanton)

For Veterans

Turn up the Creativity

The Mathematics of Various Entertaining Subjects: Research in Recreational Math (Jennifer Beineke, Jason Rosenhouse)

A fun three-volume collection of all kinds of mathematical areas from logic, graph theory and topology to music, Legos, card/board games, and much more

Proofs Without Words (Roger B. Nelsen)

Another three-volume series! This one contains only pictures to help the reader see why a particular mathematical statement (could be geometric or algebraic) may be true or how one could begin to go about proving it.

Mathematical Fallacies, Flaws, and Flimflam (Edward J. Barbeau)

This book (and its sequel) encourages readers to find out errors in the seemingly “true proofs” of false statements

Collections

(Almost) Impossible Integrals, Sums, and Series (Cornel Ioan Vălean)

For the lovers of challenging math problems. And it has a sequel too

Some more collections of specific topics followed by few general curiosities!

• Thirty-three Miniatures: Mathematical and Algorithmic Applications of Linear Algebra (Jiri Matousek)
• Exploring Continued Fractions: From the Integers to Solar Eclipses (Andrew J. Simoson)
• Curves for the Mathematically Curious: An Anthology of the Unpredictable, Historical, Beautiful, and Romantic (Julian Havil)
• Ingenuity in Mathematics (Ross Honsberger)
• Mathematical Miniatures (Svetoslav Savchev, Titu Andreescu)

100 Years of Math Milestones (Stephan Ramon Garcia, Steven J. Miller)

The centennial collection of important mathematical developments of each year from 1913-2012

The Mathematical Century: The 30 Greatest Problems of the Last 100 Years (Piergiorgio Odifreddi)

Great collection of twentieth-century mathematics from pure maths to applied maths

Fun with Numbers

Single Digits: In Praise of Small Numbers (Marc Chamberland)

All about celebrating the single digits (1 to 9) and demonstrating how they are connected to a multitude of mathematics.

The Book of Numbers (John H. Conway, Richard K. Guy)

By the legendary mathematician John Conway, this book is a journey through the world of numbers

The Biggest Number in the World (David Darling, Agnijo Banerjee)

What’s the biggest number you can think of? Check out this book to uncover exciting stories of ridiculously big numbers

Besides all that, there are also books that focus on just one number like, \(\pi, e, \sqrt{-1}, \gamma, \zeta(3)\) or a certain kind of number like the Primes or the Fibonaccis, all packed with awesome facts!

For the unaware,

• The Euler–Mascheroni constant
  \(\displaystyle\gamma =\lim _{n\to \infty }\left({\frac {1}{1}}+{\frac {1}{2}}+\cdots +{\frac {1}{n}}-\log n\right)=\lim _{n\to \infty }\left(\sum _{k=1}^{n}{\frac {1}{k}}-\log n\right)\) and
• The Apéry’s constant
  \(\displaystyle\zeta (3)=\lim _{n\to \infty }\left({\frac {1}{1^{3}}}+{\frac {1}{2^{3}}}+\cdots +{\frac {1}{n^{3}}}\right)=\lim _{n\to \infty }\left(\sum _{k=1}^{n}{\frac {1}{k^3}}\right)\)

Invitations

These are some of the books that invite people into mathematical excursions of different fields of mathematics, either straightforwardly or through their applications of significant importance to the world.

I especially love the last two books, which discuss impossible problems that humanity has struggled to solve for hundreds or even thousands of years.

• An Invitation to Mathematics: From Competitions to Research (Dierk Schleicher, Malte Lackmann)
• A Mathematical Odyssey: Journey from the Real to the Complex (Steven G. Krantz, Harold R. Parks)
• The New Turing Omnibus: Sixty-Six Excursions in Computer Science (A. K. Dewdney)
• In Pursuit of the Unknown: 17 Equations That Changed the World (Ian Stewart)
• Tales of Impossibility: The 2000-Year Quest to Solve the Mathematical Problems of Antiquity (David S. Richeson)
• Proven Impossible: Elementary Proofs of Profound Impossibility from Arrow, Bell, Chaitin, Gödel, Turing and More (Dan Gusfield)

Puzzles

Puzzles are a special kind of miniatures, when done right, are super fun and way more relatable for everyone, making them a great way to get people into math!

Recreational

Winning Ways for Your Mathematical Plays (Elwyn R. Berlekamp, John H. Conway, Richard K. Guy)

The four-volume classic on fun puzzles, which also lays the foundation to the mathematical approach to playing games

The Raven’s Hat: Fallen Pictures, Rising Sequences, and Other Mathematical Games (Jonas Peters, Nicolai Meinshausen)

A series of seeming unsolvable games that lead to the discovery of revolutionary mathematical concepts

And more!

• The Canterbury Puzzles (Henry E. Dudeney)
• Are You Smart Enough to Work at Google? Trick Questions, Zen-like Riddles, Insanely Difficult Puzzles, and Other Devious Interviewing Techniques You Need to Know to Get a Job (William Poundston)
• Algorithmic Puzzles (Anany Levitin, Maria Levitin)

Logic

• Can You Outsmart an Economist?: 100+ Puzzles to Train Your Brain (Steven E. Landsburg)
• The Gödelian Puzzle Book: Puzzles, Paradoxes and Proofs on Gödel’s incompleteness theorems (Raymond M. Smullyan)
• To Mock a Mockingbird and Other Logic Puzzles in Lambda Calculus: The Inside Story of Bletchley Park (Raymond M. Smullyan)
• Games for Your Mind – The History and Future of Logic Puzzles (Jason Rosenhouse)

Probability

• The Monty Hall Problem and its Variations: The Remarkable Story of Math’s Most Contentious Brain Teaser (Jason Rosenhouse)
• Duelling Idiots and Other Probability Puzzlers (Paul J. Nahin)
• Digital Dice: Computational Solutions to Practical Probability Problems (Paul J. Nahin)

Historical

There are tons of ways to dive into history of math, I think the best way to go with it is through engaging stories and not the dry, chronological reading. So, let us start with interesting collections of such stories.

Biographical

The Man Who Loved Only Numbers: The Story of Paul Erdös and the Search for Mathematical Truth (Paul Hoffman)

Remember Erdős? our nomad from first section he wandered around the world in 83 years with two suitcases and collaborated with over 500 researchers. Such prolific output led to the creation of the Erdős number: a measure of the “collaborative distance” between a mathematician and Erdős, based on co-authored papers. Calculate your Erdős number now!

Euler: The Master of Us All (William Dunham)

Experience the world of math through the works of the GOAT polymath Leonhard Euler

Alan Turing: The Enigma (Andrew Hodges)

The life-story of the father of theoretical computer science and his influence during World War II

A Beautiful Math: John Nash, Game Theory, and the Modern Quest for a Code of Nature (Tom Siegfried)

The legacy of an Abel Prize winner

The Story of Proof: Logic and the History of Mathematics (John Stillwell)

An unconventional biography, not of an individual, but of a concept, tracing its significant historical milestones in the development of mathematics

An Episodic History of Mathematics: Mathematical Culture through Problem Solving (Steven G. Krantz)

No time to read full biographies? Here’s a collection of short and sweet stories of important figures in math

And More Miniatures

• Sherlock Holmes in Babylon and Other Tales of Mathematical History (Marlow Anderson, Victor Katz, Robin Wilson)
• Who Gave You the Epsilon? And Other Tales of Mathematical History (Marlow Anderson, Victor Katz, Robin Wilson)

But if you do have time to spare, try the monographs of stories of some groundbreaking formulas and results in mathematics.

Monographs

The Metaphysics of the Pythagorean Theorem: Thales, Pythagoras, Engineering, Diagrams, and the Construction of the Cosmos Out of Right Triangles (Robert Hahn)

Brings together the philosophy and geometry of the Greeks

The Secret Formula: How a Mathematical Duel Inflamed Renaissance Italy and Uncovered the Cubic Equation (Fabio Toscano)

The roots of a quadratic polynomial \(ax^2 + bx + c\) can be written as \(\displaystyle r=\frac{-b\pm\sqrt{b^2-4ac}}{2a}\) or equivalently as \(\displaystyle\frac{-2c}{b\pm\sqrt{b^2-4ac}}\), so, what is the formula for the roots of this cubic \(ax^3+bx^2+cx+d\)?

Abel’s Proof: An Essay on the Sources and Meaning of Mathematical Unsolvability (Peter Pesic)

Is there a formula to solve quartic equations \((ax^4+bx^3+cx^2+dx+e=0)\)? Yup! But for quintic equations \((ax^5+bx^4+cx^3+dx^2+ex+f=0)\)?

Dr. Euler’s Fabulous Formula: Cures Many Mathematical Ills (Paul J. Nahin)

The book about the most beautiful equation is math \(\displaystyle e^{i\pi}+1=0\)

Euler’s Gem: The Polyhedron Formula and the Birth of Topology (David S. Richeson)

For any polyhedra, its \(\#\text{vertices}-\#\text{edges}+\#\text{faces}\), i.e., \(V-E+F\), is a constant called the Euler characteristic \(\chi\), which has deep connections to modern geometry

Fermat’s Enigma: The Epic Quest to Solve the World’s Greatest Mathematical Problem (Simon Singh)

I have discovered a truly marvellous description of this book, but this blog is already too long to contain

The Riemann Hypothesis: A Million Dollar Problem (Roland van der Veen, Jan van de Craats)

One of the Millennium Prize Problems, solving it is considered as the hardest way to earn a million dollars

The Golden Ticket: P, NP, and the Search for the Impossible (Lance Fortnow)

Another one of the Millennium Prize Problems, whose result can have stunning practical consequences

In Pursuit of the Traveling Salesman: Mathematics at the Limits of Computation (William J. Cook)

A seemingly simple problem of graph traversal while visiting each vertex exactly once, holds the key to solve the P versus NP problem

Four Colors Suffice: How the Map Problem Was Solved (Robin Wilson)

Another problem of graph theory with one of the most controversial proofs of mathematics

Unusual Textbooks

And lastly, here are some books that effectively teach essential topics while disguising itself as an accessible mathematical gallery filled with interesting concepts and tales.

Discrete Math

• Concrete (CONTinuous + disCRETE) Mathematics: A Foundation for Computer Science (Ronald Graham, Donald Knuth, Oren Patashnik)
• Permutation Puzzles: A Mathematical Perspective (Jamie Mulholland)
• Taking Sudoku Seriously: The Math Behind the World’s Most Popular Pencil Puzzle (Jason Rosenhouse, Laura Taalman)
• Summing It Up: From One Plus One to Modern Number Theory (Avner Ash, Robert Gross)
• Proofs that Really Count: The Art of Combinatorial Proofs (Arthur T. Benjamin, Jennifer J. Quinn)
• Generating Functionology (Herbert S. Wilf)
• The Joy of Abstraction: An Exploration of Math, Category Theory, and Life (Eugenia Cheng)

Calculus

• Infinite Powers: How Calculus Reveals the Secrets of the Universe (Steven H. Strogatz)
• When Least Is Best: How Mathematicians Discovered Many Clever Ways to Make Things as Small (or as Large) as Possible (Paul J. Nahin)

Computer Stuff

• Code: The Hidden Language of Computer Hardware and Software (Charles Petzold)
• The Logician and the Engineer: How George Boole and Claude Shannon Created the Information Age (Paul J. Nahin)
• Hacker’s Delight: An Arithmetic Algorithm Book (Henry S. Warren, Jr.)
• The Code Book: The Secret History of Codes and Code-breaking (Simon Singh)

Miscellaneous

• Probability Tales: Explores Real-World through the Lens of Probability Theory (Charles M. Grinstead, William P. Peterson, J. Laurie Snell)
• The Shape of Space: Understanding Geometry and Topology (Jeffrey R. Weeks)

Conclusion

And that’s it from my side! Hope you enjoyed reading this and found some interesting books to check out!