# Unconventional mathematics books

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:

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.

• Louis Kauffman's book "On knots." unalmed.edu.co/~mmtoro/doc/nudos/kauffman.pdf Page through it a bit to see what I mean. – Cheerful Parsnip Oct 23 '15 at 23:26
• @GrumpyParsnip Nice to see a link from my university, heheheh. – Miguelgondu Oct 29 '15 at 3:10
• Are these pictures from the recent Werner Oechslin reproduction, or are these from the an original 1847 edition? – alex.jordan Nov 1 '15 at 19:22
• @alex.jordan I believe they are from the original 1847 edition. – seeker Nov 1 '15 at 19:27
• Are you interested in crackpot mathematics books too, in that they are unorthodox? I have a copy of Heisel's book about squaring the circle. It's very difficult to read, since it stops making sense at about page 1. But there is something beautiful about his diagrams. And you can feel in his words how hurt he feels by the scorn that mathematicians have given him. – alex.jordan Nov 1 '15 at 19:30

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.

• Linderholm's book mentions the fictitious "University of Both Putfords". Am I wrong to think that sounds somehow even more English than "Oxford" sounds? Like the way I feel sure there's a town in Canada called "Mooseneck" even though I know there's not. The dedication to Clement Durell, an author of mathematics textbooks, alludes to modal logic by saying "without whom this book would not have been necessary". ${}\qquad{}$ – Michael Hardy Oct 26 '15 at 0:31
• This is truly a bizarre yet wonderful book. – rogerl Oct 29 '15 at 20:37

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

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.

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.

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.

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.

• @seeker: Thanks a lot for granting the bounty! Best regards, – Markus Scheuer Nov 2 '15 at 7:20
• +1 For the book by Dieudonné; it made a huge impression on me as a student. In contrast, the preface to Postmodern Analysis by Jost gives a radically different opinion more than 40 years later. – dafinguzman Nov 17 '15 at 3:52
• @dafinguzman: Thanks for your nice comment. I was also deeply impressed, when I was a student! :-) I'm curious, I will have a look at your reference. – Markus Scheuer Nov 17 '15 at 7:36
• It certainly is interesting to observe how the concept of what mathematics is about changes throughout the years and fashions. It's always similar, of course, but with different flavours. – dafinguzman Nov 17 '15 at 17:28

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.

• Jaynes is annoying sometimes, when he misunderstands those who disagree with him and when he addresses philosophical questions that he doesn't understand. He thought that mathematicians disapprove of the use of Dirac's delta function and other generalized functions. He was brilliant. I think on the web you can find a lot of his stuff not only on physics and probability, but also on music.When the grand piano he had shipped from Austria arrived, he unpacked it from its wooden crate and spent the rest of the day using the high-quality wood from the crate to build${}\,\ldots\qquad{}$ – Michael Hardy Oct 26 '15 at 0:36
• $\ldots\,{}$a new gate in his garden. It was a physical workout for his hands, so he was not surprised to find he could not play some loud compositions by Beethoven as well as he usually could, but he was surprised to find he could do better than usual with very quiet pieces. He wondered why, and wrote a paper about the thermodynamics of muscles, explaining it. Maybe that's on the web somewhere too. ${}\qquad{}$ – Michael Hardy Oct 26 '15 at 0:38
• Let the outright dismissal of any measure theory be unusual, the fact that Jaynes does count as a relevant statistician is confirmation of my own prejudice against measure theory and Lebesgue integration; I don't comprehend where it is good for. – Han de Bruijn Oct 27 '15 at 15:57

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.

• Seconded, especially Lakatos. – marty cohen Oct 23 '15 at 23:44
• The book Calculus by L.V, Terasov is also written in the form of a Dialogue – Kartik Oct 24 '15 at 12:13

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.

• Spencer Brown's interpretation of propositional logic is actually simple to summarize. It is equivalent with the following. We have two symbols: $($ and $)$. Let the empty string be a formula and let the parentheses occur in pairs: for example $\;(()(()()(())(()))\;$ is a formula. Then we have the following two rules of interference. And that's all: $$(()) = \qquad ()() = ()$$ – Han de Bruijn Oct 27 '15 at 15:45
• @HandeBruijn, What are your thoughts on the book? What is its general reception among mathematicians? Alan Kay mentioned it as being popular among the folks at PARC in the 70s... – Jonah Jun 2 '16 at 14:08
• @Jonah: A theory that is so simple to summarize shouldn't fill a whole book, I think. But it's all quite some time ago that I've read it and I don't have it on my bookshelves. – Han de Bruijn Jun 2 '16 at 15:55

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:

A few others:

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

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).

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.

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

• Could you include a little about why you're recommending this book? Thanks – seeker Oct 23 '15 at 23:40
• I read it many years ago, and enjoyed its presentation of the way many topics developed. Take a look inside the pages. – marty cohen Oct 23 '15 at 23:44
• Mumford wrote approvingly of this book in his Notices article "Calculus Reform--For the Millions." – Potato Oct 24 '15 at 0:24
• It's a very nice book indeed, but I'm not sure how "unconventional" I'd consider it. Unless, perhaps, being a clear and readable introduction to mathematics counts as "unconventional", which, alas, may well be the case. – Ilmari Karonen Oct 24 '15 at 9:53
• I like it enough that I am always looking for a chance to recommend it, even in an unconventional place like this. – marty cohen Oct 24 '15 at 17:32

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.

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:

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

Here's my list.

(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.

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.

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

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.

Wenzel Jamnitzer, Perspectiva Corporum Regularium

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

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. =)