Math Book List 2018 Edmund Harriss (@gelada) 1

We are truly living in a golden age of mathematics books that sneak out of the ivory tower, but this can only make it harder to find the right one. There are also quite a few out there of mediocre or even poor quality. I hope this list helps narrow down the options. Each book has comments so see what piques your fancy or just choose at random, there are no duds here.

Before you do that, however, there is one thing I would recommend to all:

Martin Gardner's Books http://www.amazon.com/Martin-Gardner/e/B000AP8X8G/

Martin Gardner never studied math at college (he majored in philosophy) but did more to stimulate an appreciation for deep and beautiful mathematical ideas in the US then all mathematics professionals combined. His books are full of wonderful non-technical mathematics presented with contagious enthusiasm and love.

Edmund’s Materials

Patterns of the Universe/Snowflake Seashell Star https://www.amazon.com/Patterns-Universe-Coloring-Adventure-Beauty/dp/1615193235 Visions of the Universe/Visions of Numberland https://www.amazon.com/Visions-Universe-Coloring-Journey-Mysteries/dp/1615193677/ US/UK titles, two colouring books full of mathematical images to colour in, if you want to learn as well that is entirely up to you!

Curvahedra https://curvahedra.com A construction system of pieces to make surfaces, explore a lot of different topics in mathematics, especially geometry without really thinking about it.

! Initially developed following Epsilon Camp 2014 with input and contributions from Rolfe Schmidt, Tara 1Smith, Chaim Goodman Strauss, Arkady Vaintrob and Harold Reiter.

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Puzzles and Fun

Can you solve my problems? by Alex Bellos https://www.amazon.com/Can-You-Solve-Problems-Perplexing/dp/161519388X Not just a collection of puzzles and problems but the history and stories around them as well.

The Number Devil by Hans Magnus Enzensberger http://www.amazon.com/Number-Devil-Mathematical-Adventure/dp/0805062998/ For pure “fun”!

The Man who Counted by Malba Tahan http://www.amazon.com/Man-Who-Counted-Collection-Mathematical/dp/0393309347/ Another fun book around Math

Things to make and do in the Fourth Dimension by Matt Parker https://www.amazon.com/Things-Make-Fourth-Dimension-Mathematicians/dp/0374535639/ Matt Parker, a mathematician who also works as a standup comic leads people through a variety of activities, games and ideas from number theory and graph theory to the 4d geometry of the title.

This is not a math book by Anna Weltman https://www.amazon.com/This-Not-Maths-Book-Activity/dp/1782402055 This is not another math book by Anna Weltman https://www.amazon.com/This-Not-Another-Maths-Book/dp/1782404988 Two great activity books exploring simple ways to make art with math, yes despite their name you will learn a lot of math, though it will be a lot of fun on the journey!

The game SET http://www.setgame.com/ Includes many educational resources associated with the game SET, a fun game with a deep mathematical background.

Doodling in Math Class (and other videos) https://www.youtube.com/user/Vihart/ Engaging and funny videos about what happens when a mind wanders in math class, the results are great doodle games that might not be quite as random as they appear.

The Big Book of Brain Games by Ivan Moscovich. http://www.amazon.com/Big-Book-Brain-Games-Mathematics/dp/0761134662/ Not just any "brain games" here, the challenges in this book have real mathematical content. If you're looking for a good source of problems that you'll have to wrestle with to solve, this book is a great alternative to competition Math.

The Moscow Puzzles: 359 Mathematical Recreations by B. Kordemsky http://www.amazon.com/Moscow-Puzzles-Mathematical-Recreations-Recreational/dp/0486270785/ An excellent collection of fun math puzzles of various difficulty for all ages.

History and Culture

*Mathematics, A Very Short Introduction by Tim Gowers http://www.amazon.com/Mathematics-Short-Introduction-Timothy-Gowers/dp/0192853619/ Fitting the title this is a short, elegant book about mathematics and mathematical thought. The author Tim Gowers is a first rate mathematician (Fields Medal winner no less) but is also very strong at communication, so this is both a clear presentation of ideas and comes from the top of the subject. I give this to anyone who asks about what I do.

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What Is Mathematics? (An Elementary Approach to Ideas and Methods) by R Courant and H Robbins http://www.amazon.com/Mathematics-Elementary-Approach-Ideas-Methods/dp/0195105192/ A great 1941 classic, still without rivals. It presents fundamental mathematical ideas through carefully selected examples from number theory, geometry, topology and calculus. Many mathematicians credit this book with opening their eyes and changing their lives.

Googols, Fractals, and Other Mathematical tales by Theoni Pappas. http://www.amazon.com/Fractals-Googols-Other-Mathematical-Tales/dp/0933174896/ Offers a fun tour of some important ideas and problems in Mathematics. Theoni Pappas has written quite a few books, and they are all worth a look.

Here’s looking at Euclid by Alex Bellos http://www.amazon.com/Heres-Looking-Euclid-Counting-Awe-Inspiring/dp/1416588280/ The Grapes of Math by Alex Bellos http://www.amazon.com/Grapes-Math-Reflects-Numbers-Reflect/dp/1451640099/ Two books that describe the culture of mathematics and mathematics in culture, from tribes in the Amazon with only the words one, two, few, many for number and abacus schools to catenary curves and Cellular automata, even the strange world of people’s favorite number.

Mathematics: From the Birth of Numbers by Jan Gullberg http://www.amazon.com/Mathematics-Birth-Numbers-Jan-Gullberg/dp/039304002X/ The book is unique among popular books on mathematics in combining an engaging, easy-to-read history of the subject with a comprehensive, profusely illustrated survey text. Intended, in the author's words, "for the benefit of those who never studied the subject, those who think they have forgotten what they once learned, or those with a sincere desire for more knowledge," it links mathematics to the humanities, linguistics, the natural sciences, and technology.

Logicomix by Apostolos Doxiadis and Christos Papadimitriou https://www.amazon.com/Logicomix-search-truth-Apostolos-Doxiadis/dp/1596914521 A comic book treatment of the story of Russel’s search for strong foundations of mathematics. The mathematics is dealt with very well and the history is reasonable (especially if you read the notes at the end). A great way to see the humans behind mathematics and the strange ways the ideas accumulate.

The Math Book by Clifford A. Pickover. http://www.amazon.com/Math-Book-Pythagoras-Milestones-Mathematics/dp/1402788290/ This offers interesting 1-page descriptions of 250 important milestones in the development of Mathematics. On its own it provides a nice perspective on the growth and breadth of the field, but it also serves as a nice “shopping list” for ideas you might want to learn more about.

Proof and Logic

Raymond Smullyan's Books http://www.amazon.com/Raymond-M.-Smullyan/e/B000AQ1NF0/ Raymond Smullyan's puzzle books, such as The Lady or The Tiger, This Book has no Title, or What is the Name of This Book (and several others) Logical Labyrinths is a good next step for many after seeing the initial logic puzzles in Martin Gardner.

Conjecture and Proof by Miklos Laczkovich. http://www.amazon.com/Conjecture-Proof-Classroom-Resource-Materials/dp/0883857227/ This book offers mini-introductions to various areas of Mathematics including topics in Geometry, Topology, and Number Theory. Along the way, it provides accessible but rigorous proofs of some deep results.

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Surreal Numbers by Donald Knuth http://www.amazon.com/Surreal-Numbers-Donald-E-Knuth/dp/0201038129/. Follow a young couple escaping from the world on a beach in India as they explore new worlds of numbers and learn what being a mathematician means.

Roger Nelson’s Books http://www.amazon.com/Roger-B.-Nelsen/e/B001IOHABO/ Nice proofs of elementary results and would be accessible to stronger students.

Proofs from the book by M Aigner, G M Ziegler and K H Hofmann http://www.amazon.com/Proofs-BOOK-Martin-Aigner/dp/3642008550/ Erdös said that God has a Book of the best (most elegant or inspiring) theorem for each problem. This volume contains some of them. (Possibly wait for an upcoming newer edition.)

Problem Solving

Crossing the River with Dogs by Ted Herr, Ken Johnson and Dan Piraro http://www.amazon.com/Problem-Solving-Strategies-Mathematical-Adventures/dp/1559533706/ This is a nice introduction to problem solving.

For all Practical Purposes (COMAP) http://www.amazon.com/All-Practical-Purposes-Mathematical-Literacy/dp/1429254823/ This provides an accessible introduction to a number of topics not part of the standard curriculum (graph theory, voting schemes, and so forth).

Mathematical Circle Diaries, Year 1: Complete Curriculum for Grades 5 to 7 by Anna Burago http://www.amazon.com/Mathematical-Circle-Diaries-Year-Curriculum/dp/0821887459/ The book contains detailed well-organized math circle lessons with problems, solutions and pedagogical advice, enough for a group meeting once a week to occupy themselves for a whole school year.

Mathematical Circles: Russian Experience by D.Fomin, S.Genkin, I.Itenberg http://www.amazon.com/Mathematical-Circles-Russian-Experience-World/dp/0821804308/ This book offers wonderful series of progressively more challenging problems in number theory, combinatorics, graph theory, and more.

Math Olympiad Contest Problems for Elementary and Middle Schools by George Lenchner. http://www.amazon.com/Olympiad-Contest-Problems-Elementary-Schools/dp/0962666211/ These books have problems (with solutions) for past MOEMS contests. The problems are a great way to develop logical and algebraic thinking without demanding significant prerequisites.

Creative Problem Solving in School Mathematics by George Lenchner. http://www.amazon.com/Creative-Problem-Solving-School-Mathematics/dp/1882144104/ Not a very well organized book, but it offers a toolkit of general problem solving strategies, a tour of different common problem types, and a nice collection of practice problems.

The Art and Craft of Problem Solving by Paul Zeitz http://www.amazon.com/Art-Craft-Problem-Solving/dp/0471789011/ An excellent book both for experienced problem-solvers and serious beginners. It contains many well-selected problems (most without solutions though).

Geometry, Origami and Topology

Crocheting Adventures in Hyperbolic Planes by D Taimina http://www.amazon.com/Crocheting-Adventures-Hyperbolic-Planes-Taimina/dp/1568814526 A satisfyingly hands on approach to Hyperbolic Geometry.

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Euler’s Gem: The Polyhedron Formula and the Birth of Topology by D Richeson http://www.amazon.com/Eulers-Gem-Polyhedron-Formula-Topology/dp/0691154570/ A readable and fast paced account of the history, and central ideas of topology, built around the great insight of the importance of the Euler Characteristic.

Which one doesn’t belong by Christopher Danielson https://www.amazon.com/gp/product/1625310811/ A book with one question where every answer is right. A fantastic way to start thinking about the justification rather than correctness and play with ideas of which things are the same or different. Also check out “How Many?” Christopher’s other marvellous books.

Geometry Revisited by H.S.M. Coxeter, S.L.Greitzer http://www.amazon.com/Geometry-Revisited-Mathematical-Association-Textbooks/dp/0883856190/ A very good textbook on "advanced" plane geometry, a perfect continuation for a motivated student after a standard high school course. It contains many beautiful and nontrivial theorems (e.g. those of Ceva, Menelaus, Pappus, Desargues, Ptolemy, Pascal, Brianchon, and Morley) and emphasizes the use of transformations (including inversions and projective transformations).

Ruler and Compass Games There are many great compass and straightedge construction "games" for example http://sciencevsmagic.net, http://euclidthegame.com and https://www.euclidea.xyz/

Zome Geometry by George W. Hart and Henri Piccioto. http://www.amazon.com/Zome-Geometry-Hands--Learning-Models/dp/1559533854/ This book provides a series of hands-on activities combined with pencil-and-paper questions that will give a solid understanding of some fundamental results in 2 and 3D geometry (with some 4D stuff thrown in). As a bonus, the lessons give the reader a solid understanding of how to use the Zome system.

How to Fold It by Joseph O’Rourke. http://www.amazon.com/How-Fold-It-Mathematics-Polyhedra/dp/0521145473/ An accessible but still serious book about the mathematics of origami, linkages, and more.

Polyhedron Models by Magnus J. Wenninger. http://www.amazon.com/Polyhedron-Models-Magnus-J-Wenninger/dp/0521069173/ With plenty of pictures of complex polyhedra, this is a fun book to flip through. If you start reading, it gets even better as you start to understand the subtleties of what you’re looking at. You might even find yourself cutting up some paper and building them yourself.

The 59 Icosahedra by H S M Coxeter, Du Val, Flather, and Petrie. http://www.amazon.com/Fifty-Nine-Icosahedra-H-S-Coxeter/dp/1907550089/ More focused than “Polyhedron Models”, this book is still full of interesting pictures, but also offers a great example of systematic analysis of a geometric problem.

Project Origami by Thomas Hull http://www.amazon.com/Project-Origami-Activities-Exploring-Mathematics/dp/1466567910/ A variety of paper folding activities in many branches of mathematics, including curvature, topology, number theory, and combinatorics. Inspired by traditional origami and will appeal to origami fans, but does not give traditional origami instructions.

Geometry and the Imagination by David Hilbert and S. Cohn-Vossen http://www.amazon.com/Geometry-Imagination-CHEL-Chelsea-Publishing/dp/0821819984/ Another masterpiece of mathematical exposition written by one of the leading mathematicians of the 20th century. The book describes beautiful ideas and examples from various topics in Geometry (projective geometry, conic sections, regular polyhedra in 3 and 4 dimensions, lattices and crystallographic groups, non-Euclidean geometries, topology of surfaces, Gaussian curvature, etc).

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The Knot Book by Colin Adams http://www.amazon.com/Knot-Book-Colin-Adams/dp/0821836781/ A very good, very readable introduction to Knot Theory, which is closely entwined with the study of three-manifolds.

Experiments in Topology by Barr http://www.amazon.com/Experiments-Topology-Dover-Books-Mathematics/dp/0486259331/ This has some paper constructions for exploring surfaces.

Three Manifold Topology and Geometry Vol 1 by Field's Medalist William Thurston (the only volume, sadly) This will be way over their heads, but some of the students will find it stimulating in any case, and it really shows off where this subject goes.

Other Mathematics

Algorithms Unplugged by Various http://www.amazon.com/Algorithms-Unplugged-Berthold-Vöcking/dp/3642153275/ Algorithms is generally regarded as a discrete branch of mathematics---algorithms are proven to be correct and efficient, or an invariant is found for a problem to prove that there can be no such algorithm. This book gives a sampling from many algorithmic topics. Written colloquially for high school students, it has many elegant proofs and no dependence on programming languages or computers (though there are some pointers to computer-based followups on the web).

Calculus (4th edition) by Michael Spivak http://www.amazon.com/Calculus-4th-Michael-Spivak/dp/0914098918 The canonical choice of a Calculus textbook for any bright mathematically-minded (high school or college) student. Written in a clear and entertaining style with a multitude of carefully selected exercises, this book presents Calculus from scratch, with numerous (often non-trivial) applications.

Concrete Mathematics: A Foundation for Computer Science by R.L.Graham, D.E.Knuth and O.Patashnik http://www.amazon.com/Concrete-Mathematics-Foundation-Computer-Science/dp/0201558025/ A remarkable textbook on Discrete Mathematics (understood here as math needed for theoretical computer science: recurrence, sums, number theory, combinatorics, generating functions, etc) with a very good selection of material and tons of concrete examples and solved problems. Beautifully written in a clear fun-spirited style with elegant proofs, it is a serious book which requires serious thinking.

Desmos http://www.desmos.com A thoughtfully designed and elegant graphing calculator, it takes some effort to make it give an error. Can graph, make sliders and animate giving an excellent way to explore functions and more visual mathematics. The seemingly simple system is surprisingly powerful. I have some notes here: https://maxwelldemon.com/2017/07/18/functional-drawing-at-c/

Books for parents and teachers

Becoming the math teacher you wish you’d had by Tracy Zager https://www.amazon.com/Becoming-Math-Teacher-Wish-Youd/dp/157110996X A book written for middle school math teachers, but inspiring for anyone interested in teaching and education. Tracy Zager leads you through the habits of working mathematicians and show how those practises can inspire better learning and thinking at all levels.

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This is not a test by José Wilson https://www.amazon.com/This-Not-Test-Narrative-Education/dp/1608463702 A personal memoire that speaks to the challenges facing education, especially around the issue of race. José Vilson writes about his experiences as a talented student becoming a teacher, and teacher leader, in the process addressing both the joys of teaching and the systematic problems that must be addressed.

Math from Three to Seven: The Story of a Mathematical Circle for Preschoolers by Alexander Zvonkin http://www.amazon.com/Math-Three-Seven-Mathematical-Preschoolers/dp/082186873X/ A captivating and well-written account of an attempt by a professional mathematician to conduct a math circle for children between 3 and 6. In addition to many fun math-related activities, the book is full of interesting psychological and pedagogical observations.

Table Talk Math by John Stevens https://www.amazon.com/Table-Talk-Math-Practical-Conversations/dp/1946444022 A book and website all about how to have mathematical conversations and discussions around the dinner table. Lots of chance to talk about and think about maths in a relaxed environment.

Arithmetic for Parents: A Book for Grownups about Children's Mathematics by Ron Aharoni http://www.amazon.com/Arithmetic-Parents-Grownups-Childrens-Mathematics/dp/0977985253/ This book is written by a professional mathematician who once accepted a challenge to teach in an elementary school and now wants to share his experience which turned out to be truly educational and eye-opening. The first half of the book, where Aharoni talks about the nature of mathematics, the role of abstraction, and the principles of teaching it, would be interesting not only for parents of elementary school kids.

A Mathematician's Lament by Paul Lockhart http://www.amazon.com/Mathematicians-Lament-School-Fascinating-Imaginative/dp/1934137170/ One of the best critiques of the current state of math education written with eloquence and passion. The Lamentation part continues with Exultation, where the author presents his view on what mathematics really is about (the purest art form, of course). This small book is a must read for parents, math educators and politicians.

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