Unconventional mathematics books

Question

I've recently purchased Oliver Byrne's reproduction of Euclid's Elements. It's a beautiful tome, that's rather unique in its presentation of the material as it demonstrates many of Euclid's proofs as lurid and lusciously coloured geometric figures. See below:

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So, my question is:

What are some other mathematics books that convey a topic in a manner that breaks from orthodoxy?

Now I doubt there are very many books that meet at the intersection of art and mathematics such as this, so this should not be the sole criteria by which the 'unconventionality' of a book should be judged. In all probability any departure from orthodoxy will likely manifest itself in the form of pedagogical organisation/exposition distinctions, and so this should be the predominant criteria by which you should judge a book's eligibility for recommendation. It would also be appreciated if you could provide a justification as to why you believe a given book is unconventional.

I'm sorry if this is off-topic, hopefully I can at the very least expose a few people to this lovely book.

Answer 1

I will recommend again "Mathematics for the Million" by Lancelot Hogben. $15 at Amazon.

Answer 2

Two other books that I'd also like to recommend are:

1. Imre Lakatos, Proof and Refutation
2. Alfred Renyi, Dialogues on Mathematics

Both books are quite unconventional in the sense that their presentation of material is dialogic in nature. The first is a relatively well-known book about mathematical proof and discovery, it's a narrative framed in the form of conversations between a teacher and his students. The second book is also conversational in nature, however the conversations are between luminaries of certain fields and an inquisitive lay person, Socrates, Archimedes, and Galileo feature. The book itself touches on the nature of mathematics.

Answer 3

Carl Linderholm's Mathematics Made Difficult is quite interesting. It was described by Halmos (Linderholm's PhD adviser) as a sort of "mathematical in-joke." But you'll find reviews that, while unanimously positive, are all over the map. I think that officially makes it a work of art, since the meaning of the content is truly in the eye of the beholder (whether that was intentional or not).

At any rate, it revisits elementary mathematics armed with words like "endomorphism" and is full of incredibly weird story-telling.

Despite all the incredibly weird things that happen and its confusing nature, I do think it's nice to be reminded that the germ of our sophisticated modern mathematics is absolutely contained in the "basic math" all children learn. There's also some fun-poking at how asinine things like, for example, "mixed fractions" may be, and what children are subjected to, pedagogically.

Answer 4

Probability Theory: The Logic of Science by E. T. Jaynes is the notes turned into a book after the author's death. The point of view on probability as the quantitative measure of our belief is not exactly unorthodox, but the outright dismissal of any measure theory is very unusual. Plus the book is a wonderful read.

Answer 5

Uncle Petros and Goldbach's Conjecture, by Apostolos Doxiadis. It's not technically a book about mathematics, but it is a wonderful story about how mathematicians think about problems.

Answer 6

Laws of Form by G. Spencer-Brown is nothing if not unconventional. He has some crackpot theories on the nature of psychiatry and maybe his style of writing deserves the "crackpot" epithet as well. But I suspect there's some legitimate mathematical logic. I just never took the trouble to decipher it.

Answer 7

Here are some rather unconventional books with focus on visual perception and guarantee for many interesting and amusing hours.

• Proofs without words - Exercises in visual thinking:

The title is program. Volume I and II by Roger B. Nelson follow the motto: A picture is worth a thousands words and present graphical solutions without words to rather elementary problems.

• A Topological Picturebook by George K. Francis

is definitely more challenging. It is full of wonderful pictures of topological structures. Many of them are a little masterpiece of drawing art and support this way a better understanding of the theme. But even these wonderful graphics will be surpassed by those of the following book.

• The Ashley Book of Knots by Clifford W. Ashley

Although this book is not written for mathematicians, I recommend it to all who like topology. The theme of the last chapter from the Topological Picture Book is knot theory. And if you are a visual learner with a faible for knots you will appreciate this book . It is a guide containing thousands of wonderfully drawn knots most of them are masterpieces of art. You can delve into an incredible world of different knots and after that you will look at Topology with different eyes.

• Beautiful Evidence by Edward Tufte

is a great book about presenting data and statistical information. It could be a valuable source for statisticians and those who like to think about how to improve graphical information. You will get a first impression when visiting his home-page.

Hint: The following is not a recommendation of an unconventional book but instead a perfect contrast to the books above. You may have a look at the classic Foundations of Modern Analysis from 1960 by J. Dieudonne. You will not find even one diagram or graphic in this nine volume treatise!

Dieudonne stated in his foreword:

"... This also has as a consequence the necessity of a strict adherence to axiomatic methods, with no appeal whatsoever to geometric intuition, at least in the formal proofs: a necessity which we have emphasized by deliberately abstaining from introducing any diagram in the book. My opinion is that the graduate student of today must, as soon as possible, get a thorough training in this abstract and axiomatic way of thinking, if he is ever to understand what is currently going on in mathematical research. This volume aims to help the student to build up this intuition of the abstract which is so essential in the mind of a modern mathematician..."

I deeply appreciate this classic and sometimes consult it for some valuable information. But I also have to admit, as humble mortal far from playing in the top leagues that I'm very grateful for professional graphics and excellent pictures which guide me to fruitful mathematical directions.

Answer 8

Visual thinking make me think of:

• Claudi Alsina and Roger Nelsen, When Less is More: Visualizing Basic Inequalities (2009)
• Roger B. Nelsen, Proofs without words 2. More exercises in visual thinking (2000)
• Roger B. Nelsen, Proofs without words. Exercises in visual thinking (1993)

I am a great fan of the short-story master, Jorge-Luis Borges, who played a lot on infinity, self-referencing, one-to-one mappings:

• Guillermo Martinez and Andrea G. Labinger, Borges and Mathematics (2011)

A few others:

• Marcel Berger, Geometry revealed, A Jacob's Ladder to Modern Higher Geometry (2010) is meant as a follow-up to Geometry and the Imagination, by David Hilbert and Stephan Cohn-Vossen (1932), with intuitive geometry.

• Florian Cajori, A History of Mathematical Notations (1993) provides a strong background on everyday symbols.

• J. Michael Steele, The Cauchy-Schwarz Master Class. An Introduction to the Art of Mathematical Inequalities (2004) is like peotry on a simple and basic inequality, and pushes it to upper limits.

• Marko Petkovsek and Herbert S. Wilf and Doron Zeilberger, $A=B$ (1996). Here is the foreword:

Science is what we understand well enough to explain to a computer. Art is everything else we do. During the past several years an important part of mathematics has been transformed from an Art to a Science

Answer 9

Here's my list.

• William Dunham’s Journey through Genius: The Great Theorems of Mathematics. This is a book of gems. It mixes classical mathematical results with their background stories in a balanced manner. If I remember correctly, it contains an interesting geometric proof of Heron's formula, Cardano's solution to the cubic equation, and several classical infinite sums and products.

• Martin Gardner’s My Best Mathematical and Logic Puzzles and Martin Gardner’s other books. In the former, you'll find some classical puzzles, such as the mutilated chessboard puzzle, and finding a spot on Earth, other than the north pole, such that one can walk one mile south, one mile east and one mile north and return to the original place.

• Steven Strogatz’s Nonlinear Dynamics and Chaos. This is a college-level text. But it is very readable and informative. Even to this day, I am surprised that how the author managed to explain so many deep and exciting high-level theorems and methods in such a plain and intriguing manner.

• Charles Pinter’s A Book of Abstract Algebra. This is also a textbook, written in a very clear and inviting manner. It address the mystery that polynomial equations of 5 or more degrees are not generally solvable by radicals. To be fair, reading this book requires some effort. But the learning process has been made as smooth as possible, and I believe few would complain at the end of the journey.

Answer 10

In the vein of books that meet at the intersection of art and mathematics a wonderful book that I've just discovered is: Anatolii T. Fomenko, Mathematical Impressions. The author uses detailed drawings to illustrate abstract mathematical concepts.

And obviously anything by M. C. Escher.

Answer 11

One of my favourite books is

The Fascination of Groups, by F.$~$J.$~$Budden.

This is certainly no ordinary introduction to group theory. Here's a sample from the preface, to give you a taste of the writing style:

It takes 545 pages to cover what would be completed in most text-books in one to two hundred pages. But that is precisely its raison d'etre - to be expansive, to examine in detail with care and thoroughness, to pause - to savour the delights of the countryside in a leisurely country stroll with ample time to study the wild life, rather than to plunge from definition to theorem to corollary to next theorem in the feverish haste of a cross-country run.

And then later in the same paragraph, an explanation of the contents:

The objective is to provide a wealth of illustration and examples of situations in which groups may be found and to examine their properties in detail, and the development of the elementary theory in the light of these widely ranged examples.

As promised, while the book does also work as a textbook of-sorts, giving good explanations of the definitions and theorems, and including exercises for every chapter (some of which are decidedly nontrivial, and are sometimes inserted before the necessary material to answer the question is covered, just to get the reader thinking about the topic), the true value of the book lies in its extensive collections of examples of groups, and the book is positively overflowing with illustrations and Cayley tables, all neatly organized into the relevant chapters. To top it all off, towards the end are dedicated chapters on the applications of group theory to music, campanology, geometry and patterns (in the sense of wallpaper patterns).

Answer 12

When the fact is on unconventional mathematics, the first book that appears in my mind is Magical Mathematics of Persi Diaconis and Ron Graham.

It is a book on card magics as well as rigorous mathematics, it is not simply basic combinatorics, but even the application of famous Fermat's Last Theorem in seemingly simple magic tricks! Gilbreath's Principle to Mandelbrot Set, every topic is stated in this book.

By the word 'magical Mathematics', everyone thinks that it will be childish number tricks, but the tricks stated in this book was widely acclaimed by amateur spectators to members of AMS.

Martin Gardner, who has many published books on the same topic from Dover Publication, Mathematical Association of America and also American Mathematical Society, wrote in the foreword of this book:

You can also download the e-book from here.

Believe me, this is probably the best books I have read on recreational mathematics.

Answer 13

(1) A Mathematician's Miscellany, by J.E Littlewood.A classic. Very interesting, very entertaining. (2) A Budget Of Trisectors, by Underwood Dudley. An exposition of a modern mathematician's interactions with assorted trisectors, circle-squarers,and assorted pseudo-mathematical oddballs.

Answer 14

Joe Roberts, Elementary Number Theory – the unconventional thing about this book is that it's done in calligraphy.

Answer 15

Donald Knuth's book, Surreal Numbers, an exposition of John Conway's work, is unconventional in that it is written in the form of a novel.

Answer 16

Trolling Euclid by Tom Wright is my favorite book about unsolved problems. Very nice read.

Answer 17

Reuben Hersh' 18 Unconventional Essays in the Nature of Mathematics is a (unconventional) discussion of several topics in metamathematics. You can easily find a (pre)view and have a look online.

I found this thread while looking for a pen&paper-free math book to read, and just thought that ---if I didn't know it yet--- I would love to find here this book whose reading I remember to enjoy so much. =)

Answer 18

It's not a book, but the PhD dissertation of Piper Alexis Harron (Princeton, 2016), titled "The Equidistribution of Lattice Shapes of Rings of Integers of Cubic, Quartic, and Quintic Number Fields: an Artist’s Rendering", is a uniquely subversive take on rigorous mathematics. It's a delightful read, especially if you've ever felt like pure mathematics was a bit...stuffy.

Answer 19

Wenzel Jamnitzer, Perspectiva Corporum Regularium