Math Games

Beautiful References

Ed Pegg Jr., January 4, 2006

"Gentle Reader: This is a handbook about TeX, a new typesetting system intended for the creation of beautiful books — and especially for books that contain a lot of mathematics." (Donald Knuth, Preface for The TeXBook.)

At the start of the last century, typography required long hours of work with either lead blocks or gas-powered linotype machines. The average life expectancy of a printer was 28 years. About half of them contracted tuberculosis. Even as late as the 1970's, the Environmental Protection Agency had to increase the acceptable amount of lead in a worker's bloodstream due to every Linotype operator exceeding the suggested limit. The 1970's also marked a huge change in printing as publishers replaced linotype experts with computer typography.

Unfortunately, for mathematics, the new printing methods looked terrible. Donald Knuth took a look at the galleys of The Art of Computer Programming and decided they were unpublishable. He figured he could fix computer typography in about six months. The actual time was ten years, but his creation of TeX allowed mathematical books to be beautiful again.

For this month's column, I thought I'd look at some mathematical references that I've found particularly useful — the books on my closest shelf that I wind up using regularly. I wear a lot of mathematical hats, and need references to answer dozens of questions every day. Typically, I can find a reference which has the definitive answer, and that will settle things. I'll start with the first beautiful reference of the modern era.

Reference 1: The Art of Computer Programming by Donald Knuth.

My favorite volume here is vol. 2, Seminumerical Algorithms, with lots of great info on random numbers, how computer numerics work for addition and multiplication, different number systems, factorization, and polynomials. The sorting algorithms of vol. 3 are also fascinating. Knuth's books are meticulously indexed and famously accurate (if you can find an error, Knuth will pay you $2.56). TAoCP vol 4 Fascicles 2, 3, and 4 are also available.
    Recommended for: Algorithm designers, math historians.
     Hardcover, 650 / 762 / 780 pages, Addison-Wesley, 1998, 9.5×6.6×1.6 inches.

That's not to say that all good references are modern. Dover Publications regularly brings back the classic works of the past at reasonable prices. Such Dover titles as Amusements in Mathematics (Dudeney), Pearls in Graph Theory (Hartsfield and Ringel), and The Four-Color Problem (Saaty and Kainen) are all marvelous, inexpensive books, but I don't use them very much. Often, the indices of these older books aren't the best, and I have to find the index amongst the usually outdated Dover catalog. Hopefully, Dover will one day offer modern indices for older classics. One set of books stands above the others as a beautiful, valuable reference.

Reference 2: The History of the Theory of Numbers, by Leonard Dickson.

Originally published in 1919-1923, these three books cover almost every number theory idea from the beginning of mathematics until 1920. Vol 1, Divisibility and Primality, covers prime numbers, Farey sequences, perfect numbers, Fermat numbers, and much more. Vol 2, Diophantine Analysis, covers right triangles, sums of powers, and Fermat's last theorem. Vol 3, Quadratic and Higher Forms, covers exactly that. Each page lists roughly five interesting facts, all referenced by footnotes.
   Recommended for: Amateur mathematicians, math historians.
    Paperback, 486 / 803 / 313 pages, Dover, 2005, 8.5 x 5.5 x 1.7 inches.

The reference I use the most is the website MathWorld. Full disclosure: I'm now the associate editor. Eric originally designed his Treasure Troves to be a website with occasional printed snapshots. CRC was interested, and printed the first book. Unfortunately, CRC then shut MathWorld down, which led to immediately to the creation of PlanetMath (a good site for long proofs), and spurred greater development on Wikipedia. More than a year later, the courts allowed MathWorld to return. I am personally boycotting all CRC books until they make amends. The MathWorld website is much better than the book, but I'll still recommend it:

Reference 3: Concise Encyclopedia of Mathematics, by Eric Weisstein.

An encyclopedia on everything in mathematics. Back in 1998, I was drawn to Eric's work by his commitment to making good diagrams of recreational math concepts. I soon started contributing things, like pentagon tiling and snarks. More lazily, whenever there was a concept I wanted to mention on a webpage but not explain in detail, such as Heron triangles, I would just make a MathWorld link. The book itself, particularly the first edition, is beautiful to leaf through. Do feel free to join the contributors. If you use the browser Firefox, there is a plugin for MathWorld searches, and also for Wolfram Functions searches.
   Recommended for: People that cannot use the MathWorld website.
   Hardcover, 3242 pages, Chapman & Hall/CRC, 2002, 11.4×8.5×3.2 inches.

Martin Gardner wrote the books and columns that got me started on mathematics. Back in the seventh grade, I searched through the microfiche of Scientific American back issues in the school library to find and read all his columns. The task took weeks. At the end of it, I had an unshakable love for mathematics. Now that his 25 years of columns are available on a single CD, I cannot recommend this next reference enough:

Reference 4: Mathematical Games, by Martin Gardner.

30 years of math columns from the world's foremost popular mathematics writer. This CD-rom contains all the Mathematical Games columns written by Martin Gardner (Scientific American, 1956 to 1986) on one CD in a searchable database. Type a word in the Search box, press Go and a list of columns will appear.
In these columns, Gardner eloquently describes the delights of mathematics, puzzles and problem solving. His column broke such stories as Rivest, Shamir, and Adelman on public-key cryptography, Mandelbrot on fractals, Conway on Life, and Penrose on tilings. He enlivened classic geometry and number theory and introduced readers to new areas such as combinatorics and graph theory. The search feature is fantastically useful for researchers.
   Recommended for: Everybody. Mandatory for recreational mathematicians. Excellent for young adults.
   CD-ROM PDF, 4500 pages, Mathematical Association of America, 2005, 7.8×5.0×0.8 inches

A recent addition to my nearest bookshelf has long been a classic in Europe. The Oxford Users' Guide to Mathematics is also based on the the original Bronshtein book, and I have both — I prefer the Handbook of Mathematics. I also consider it superior to the Mathematics Handbook (Råde & Westergren), the Handbook of Mathematics and Computational Science (Harris & Stocker), and the CRC Standard Mathematical Tables and Formulae (Zwillinger).

Reference 5: Handbook of Mathematics, by Bronshtein and Semendyayev

With a nice small size, high quality assembly, beautiful layout, and an extensive 50 page index, this is now the first book I turn to after MathWorld. The pages have wonderfully concise definitions and are packed with important information on each subject. Many useful diagrams help to explain the material. Note that although it is a good handbook for all areas of college math, it isn't a textbook. For example, pages 299-302 on Group theory would fill 150 pages of an introductory textbook on the subject, and Galois isn't mentioned in those pages. It's the mathematical facts — just as a handbook should be.
   Recommended for: Everybody. Useful from high school to college and beyond.
   Paperback, 1160 pages, Springer, 2003, 8.2×5.5×1.7 inches.

For a classic, dry, book with long lists of lots of functions, I generally recommend the Wolfram Functions site. The formula search is particularly useful when someone believes they have a new formula. For an actual book of function tables, Abramowitz is one of the better ones.

Reference 6: Handbook of Mathematical Functions, by Abramowitz and Stegun

This book is inexpensively priced from Dover, and is also in the public domain. Several websites feature free copies of it, such as Take a look at it (perhaps via before considering a purchase. The NIST plans to replace Abramowitz (as it is usually called) with a new version. I perhaps sound slightly hostile towards the book here, but I do wind up referencing it frequently.
   Recommended for: Function historians and Engineers.
   Paperback, 1046 pages, Dover Publications, 1974, 10.4×8.0×1.6 inches.

A high-level, nearly diagram-free reference produced by the Mathematical Society of Japan is another book I frequently pull down.

Reference 7: Encyclopedic Dictionary of Mathematics, by Kiyosi Ito.

This book aims at high level of mathematical knowledge. For example, if I looked up Lattices, I would need to find it in the index first, leading to section 243. "Definition: When x and y are elements of an ordered set L, the supremum and infimum of {x,y}, whenever they exist, are called the join and meet of x, y and denoted by x∪y and x∩y, respectively. L is called a lattice (or lattice-ordered set) when every pair of its elements has a join and a meet." (Most of the book is even more abstract.) An in-depth discussion follows, with a bibliography of classic texts at the end. An excellent set of books, but not for beginners.
   Recommended for: High-level mathematicians.
   Paperback, 2148 pages, MIT Press, 1993, 12.6×6.3×4.0 inches.

Richard Guy has helped to produce three books I use regularly. One he wrote with John Conway is The Book of Numbers, which uses an easy, friendly tone to describe scads of good mathematics.

Reference 8: The Book of Numbers, by Conway and Guy.

This book seems deceivingly simple and breezy. Lots of illustrations, gentle sentences, and very readable commentary. However, there is a lot of good math in the book, enough that I wind up referencing it frequently. The core topic of the book is numbers, in all their varieties. From a random page: "Delos is an island in the Greek archipelago, once famous as the reputed birthplace of Apollo and Artemis. The story is told that when a plague was raging at Athens, the inhabitants sent an emissary to ask the oracle of Apollo at Delos what to do. The oracle replied that the plague would cease if the altar to Apollo were exactly doubled in size." A rapt discussion of algebraic numbers follows.
   Recommended for: Students, amateur mathematicians, math historians.
   Hardcover, 310 pages, Springer, 1996, 9.5×6.4×0.8 inches.

Guy's other inescapable books involves famous unsolved problems. I also adore the Winning Ways books he wrote with Conway and Berlekamp, but I don't often get a reason to reference them.

Reference 9: Unsolved Problems in Geometry / Number Theory, by Richard Guy, with Croft & Falconer.
An interesting fraction of the mail I receive comes from people desiring a discussion of various unsolved problems. The Unsolved Problems series can usually bring these correspondents up to speed on what is known. I've been guilty of that myself— pondering some problem on my site, only to get a friendly letter from Guy to see a certain section which explains all that is known on the topic. As new revisions come out, he records the progress that has been done for all these interesting problems.
   Recommended for: Amateur mathematicians, unsolved problem solvers.
   Hardcover, 438 pages, Springer, 2004, 9.7×6.4×1.0 inches.

Michael Trott is mostly known for making lots of gorgeous images. Less known are his recently published Mathematica GuideBooks, which fully explain, extensively reference, and provide commented code for thousands of mathematical images.

Reference 10: Mathematica Guidebooks for Graphics / Numerics / Programming / Symbolics, by Michael Trott.

A truly gorgeous set of books, with about 5000 pictures, a thousand exercises, 11000 hyperlinked references, and commented code for producing everything. For each image, there is a leading discussion, and a list of related references with more information. Also, here's a small secret: if you buy any one of the books, the included DVD has all four books on it within a fully searchable set of Mathematica notebooks (updated versions in Numerics & Symbolics). For example, I looked up LATTICES, and got links to Brillouin zones of ~, charged ~, counting points in ~, cubic ~ in 3D, cubic and hexagonal ~ in 2D, morphing of ~, points visible in ~, superpositions of ~, vortex ~, ~ Green’s function and Mathematica function LatticesSuperposition. With Eric Weisstein, I wrote a column on the GuideBooks, which shows many sample graphics. The GuideBooks website contains a detailed table of contents, sample chapters, a full index, and considerable additional material. The additional material alone contains 50 detailed notebooks. To sample the additional material, or a DVD from the book itself, you can download a trial version of Mathematica, which serves as a fully functional reader after the trial expires.
   Recommended for: Mathematica users, illustrators, math graphic fans, physicists.
   Hardcover, 1340 / 1209 / 904 / 1454 pages, Springer, 2004 / 2005, 9.4×7.5×2.1 inches.

The Mathematical Constants website is much smaller than it used to be (due to a court order, sigh), but all of the richness is now available in book form.

Reference 11: Mathematical Constants, by Steven Finch.

Some of the most popular math books have been about single constants, such as e, π, i, and φ. There are hundreds of other mathematical constants. They are all lovingly described by Finch in either his book and at his supporting site. Finch has also contributed tremendously to MathWorld Constants section. Charles Ashbacher: "I consider this book to be an essential component of all mathematical libraries. I have placed it on my 'within the grasp' shelf and have strongly recommended to the college library that it be added to the reference collection." Myself, I wind up needing to look up obscure constants quite often, and always consult Finch.
   Recommended for: Anyone that likes constants.
   Hardcover, 622 pages, Cambridge University Press, 2003, 1.8×6.5×9.5 inches.

One small book does an admirable job of containing all of the must-know mathematics in one compact text.

Reference 12: Mathematical Thinking: Problem Solving and Proofs, by John D'Angelo and Douglas West.

This is a textbook, and a slightly pricey one at that. From the preface: "This book arose from discussions about the undergraduate mathematics curriculum. We asked several questions. Why do students find it difficult to write proofs? What is the role of discrete mathematics? How can the curriculum better integrate diverse topics? Perhaps most important, why don't students enjoy and appreciate mathematics as much as we might hope?" Then there is a preface for the student, with 37 classic math problems, many from the rec.puzzles FAQ. The problems serve as a springboard to all the important concepts of college math, all of which are concisely explained, along with almost a thousand exercises. When I need a definition, I often find the simplest and best definition here.
   Recommended for: Students, amateur mathematicians, math historians.
   Hardcover, 412 pages, Prentice Hall, 1999, 9.3×6.3×0.8 inches.

I regularly get asked for the history of a word. For that, one gigantic book proves worthwhile about twice weekly.

Reference 13: The Compact Oxford English Dictionary, by Weiner and Simpson.

"If there is any truth in the old Greek maxim that a large book is a great evil, English dictionaries have been steadily growing worse ever since their inception...."— From the OED Historical Introduction. It's amazing how often I need to look up the history of a particular word, and OED provides. For example, Proclus, who wrote the commentaries for Euclid's Elements, used the greek form of the word TRAPEZIUM for quadrilaterals with exactly two sides parallel, and the word TRAPEZOID for a quadrilateral with no sides parallel. In 1795, Hutton's Mathematical Dictionary accidentally swapped the definitions, and these became the standard definition in the United States and some other countries. Thus, there is no good universal definition for TRAPEZOID and TRAPEZIUM. The definition depends on the country you live in. Indeed, it depends on what a given teacher expects. I had no idea this raging 200+ year educational war even existed before looking it up in the OED.
   Recommended for: Math historians, anyone that likes good references.
   Hardcover, 2416 pages, Oxford University Press, 1991, 19.0×13.0×5.0 inches.

Here's a beautiful book that I'll actually discourage. Through the assistance of thousands of professional and amateur mathematicians, the Online Encyclopedia of Integer Sequences website has expanded vastly beyond the original book. I use OEIS every day.

Reference 14: The Encyclopedia of Integer Sequences, by Neil Sloane.

"In spite of the large number of published mathematical tables, until the appearance of the first authors A Handbook of Integer Sequences in 1974 there was no table of sequences of integers. The 1974 book remedied this situation to a certain extent, and the present work is a greatly expanded version of that book. The main table contains 5488 sequences of integers (compared with 2372 in the first book), collected from all branches of mathematics and science. The sequences are arranged in numerical order, and for each one a brief description and a reference is given. An invaluable tool. I shall say no more about this marvelous reference except that every recreational mathematician should buy a copy forthwith." — Martin Gardner in Scientific American.
   Recommended for: Fans of the OEIS website and reference collectors.
   Hardcover, 587 pages, Academic Press, 1995, 9.3×6.3×1.2 inches.

The following book has long been out of print, but it's been one of my favorites for more than a decade. It seems to be readily obtainable (Amazon currently shows 30 copies) for around $30, which is a great price for a reference of this quality.

Reference 15: VNR Concise Encyclopedia of Mathematics, by Gellert, Küstner, Hellwich, and Kästner.

An amazing book spanning almost all of mathematics. Can be described as a textbook without any exercises. In addition to the beautiful layout, color is discretely and extensively used to highlight key points. Simpler concepts, such as addition, get written at an easier level than more difficult concepts, like spherical trigonometry. Profuse illustrations are shown throughout the book. In the back, pictures of many famous mathematicians are shown.
   Recommended for: Everyone, especially students.
   Hardcover, 790 pages, Van Nostrand Reinhold, 1975, 9.4×6.6×1.9 inches.

I personally found advanced Analysis brutally difficult — I doubt I'd have gotten through the following text without lots of good instruction from my professor, Rinaldo Schinazi. Extremely difficult courses can truly be appreciated only when one is standing atop the conquered mountain.

Reference 16: Principles of Mathematical Analysis, by Walter Rudin.
This is a hard book, not very suitable for self-study. If you fully understand the book, you're a mathematician. It has a lot of excellent examples, discussions, definitions, and proofs. When I need to get the wording of an analysis definition exactly right, I'll consult Rudin first.
   Recommended for: Graduate students struggling with analysis.
   Hardcover, 342 pages, McGraw-Hill, 1976, 9.3×6.2×0.8 inches.

I frequently get asked for how to solve math problems. The following text by Engel is superb. It's now considered a mandatory text for students competing seriously in math olympiads. Another good book of this type is Mathematical Circles (Fomin).

Reference 17: Problem-Solving Strategies, by Arthur Engel.
Most math texts provide a grounding in mathematics, but not in problem solving. This book provides lots of good instruction on how to tackle seemingly impossible problems. The chapters outline the various strategies: the invariance principle, coloring proofs, the extremal principle, the box principle, enumerative combinatorics, number theory, inequalities, the induction principle, sequences, polynomials, functional equations, geometry, games, and further strategies. Many of the listed problems have deep theory behind them. For example, find Euler's proof for the following: If n ≥ 3, then 2n can be represented in the form 2n = 7 x2 + y2 with odd integers x and y. I often need to explain these same concepts.
   Recommended for: Students, amateur mathematicians, math olympians.
   Paperback, 403 pages, Springer, 1999, 9.2×6.2×0.9 inches.

I'll finish with a list of some of my favorite online references, some already mentioned.

Reference 18: Online references.

MathWorld — A huge encyclopedia of mathematics by Eric Weisstein.
History of Mathematics — Jeff Miller's "earliest usages" list.
OEIS — Over 100000 important sequences, maintained by NJA Sloane.
Free Online Math Journals — by
MacTutor History of Mathematics — Math biographies.
Free Online Math Textbooks — Maintained by Alex Stef.
Cut the Knot — by Alex Bogonolmy. Lots of good mathematics.

Functions — The Wolfram Functions site.
Geometry Junkyard — by David Eppstein.
Mathematical Constants — by Steven Finch.
Math Symbols — by
Math ArXiv — thousands of online math papers.
Math Pronunciations Guide — by Kent Kromarek
Google — For everything else.

To some extent, I hope to answer many common math questions with this column. If you are a student and have a question about a math, I recommend the team at Ask Dr. Math. For help with algebra and calculus problems, I recommend An excellent computer excursion into mathematics is Stan Wagon's The Mathematical Explorer.

Thus ends my list of beautiful references, hopefully with enough caveats that I'll lead no-one astray. A list of beautiful math books would be considerably longer. Here is one of my older book lists, for example, to which I need to add such beautiful books as Set Theory (Jech), A New Kind of Science (Wolfram), Triangle Centers and Central Triangles (Kimberling), and Puzzle Cyclopedia (Nikoli). I'm always on the look out for beautiful reference books and websites, so feel free to recommend something to me.

References (also, see above):

Gene Gable, "Heavy Metal Madness,", "Just What is TeX?", "The Linotype,"

Math Games archives.

Comments are welcome. Please send comments to Ed Pegg Jr. at

Ed Pegg Jr. is the webmaster for He works at Wolfram Research, Inc. as an associate editor of MathWorld.