An Idiosyncratic Annotated Bibliography

- Philip J. Davis & Reuben Hersh, "The Mathematical Experience," Houghton Mifflin, 1981.

A wonderful mix of mathematics, philosophy, history of mathematics, and what might be called the psychology of mathematics as currently practiced by working mathematicians. This is one of the few books that really shows a non-mathematical audience what it's like to do serious mathematics. The authors do get involved with philosophical issues most mathematicians do not get worked up about, and it's worth keeping that in mind when reading the philosophical parts. It's also worth slogging through any technical points for the overall richness of this book.

- Morris Kline, "Mathematics: A Cultural Approach," Addison- Wesley, 1962.

An outstanding book covering much of mathematics in both ancient and modern times. The book discusses historical and philosophical issues without neglecting more traditional mathematical ones. Some of the interesting chapters include 8 (Ancient Greece), 10 (Renaissance Painting), 11 (Projective Geometry) and 24 (Music). Read the rest of the book while you're at it. Kline has a similar book in a Dover edition.

- Ivars Peterson, "The Mathematical Tourist," W. H. Freeman, 1988.

A worthy exposition of a variety of mathematical topics, from prime numbers to knot polynomials to fractals (what else?). What makes this book unique is that the topics covered have all seen much progress at the research level in recent years, offering a rare chance for non-mathematicians to hear about what's happening in the mathematical world at this very moment (as opposed to most school mathematics, which is usually hundreds, if not thousands, of years old). My only complaint is that there are no exercises, so that the book does all the thinking, instead of the reader.

- James R. Newman, "The World of Mathematics," four volumes, Simon and Schuster, New York, 1956.

Before the recent explosion of books aimed at showing a general audience what really goes on in mathematics, this four volume set was one of the best places to turn -- and it still merits a look. The subtitle of the set tells it all: "A small library of the literature of mathematics from A'h-mose the Scribe to Albert Einstein, presented with commentaries and notes by James R. Newman." Among the pieces are excerpts from great mathematicians such as Descartes, Euler and Poincaré, and equally great thinkers such as Galileo, Heisenberg and Russell. Time spent with these volumes will be well spent indeed.

- H. Steinhaus, "Mathematical Snapshots," Oxford U. Press, 1983.

Newly reprinted, and well worth the wait. Steinhaus is a well known mathematician who managed to roam the mathematical landscape in one continuous journey, stopping at all kinds of interesting places. He mentions everything from beehives to musical scales to the Konigsberg bridges. What a trip! I don't know how he does it.

- David E. Penny, "Perspectives in Mathematics," W. A. Benjamin, 1972.

A wonderful book dealing with various topics in mainstream contemporary mathematics, such as dissections of polygons, groups and infinite sets. Each section contains some pretty results usually taught fairly late in a standard mathematical career. Of particular interest is the section called "The Well-Tempered Clavichord." A serious text, the reader should be on a level to take calculus, and be willing to work hard to go through thisbut the rewards are many.

- David Wells, "Hidden Connections, Double Meanings," Cambridge U. Press, 1988.

An interesting collection of explorations into various topics in mathematics, leading the reader on as much by questions, hints and puzzles as by straightforward exposition. I rarely like books based largely on puzzles, but in this one the puzzles actually lead to substantial mathematical ideas (as opposed to being concocted with the sole aim of stumping the reader). Worth a try, in spite of some rather tricky spots.

- Rudy Rucker, "Mind Tools," Houghton Mifflin, 1987.

Written in Rucker's incomparable style, the idea is to view mathematics from five levels: number, space, logic, infinity and information. The last of these gets the most attention, not surprisingly in this age of computers and the Internet. Unfortunately, Rucker's attempt to get his philosophical point across leads to a choice of topics that is less than thrilling at all times (though good in parts), at least according to my taste.

- Douglas M. Campbell & John C. Higgins, eds., "Mathematics - People, Problems, Results," Wadsworth, 1984.

A three volume collection of essays on various topics about mathematics and mathematicians. The style is often historical, and wordy, but some of the essays are quite nice. The broad range of mathematics is displayed. Among other essays, read "Mathematics as Propaganda" by Neal Koblitz in Vol. III.

- Jeffrey R. Weeks, "The Shape of Space," Marcel Dekker, 1985.

An outstandingif somewhat schizophrenicbook. It starts out being a chatty sequel to "Flatland," introducing the reader to many fundamental concepts of modern topology and geometry. I know of no other book that treats these topics in such a lucid and accessible way. Unfortunately, the book becomes fairly technical after a while, since the author is set on presenting the recent (and very important) work of William Thurston on the geometry of 3-manifolds. Coincidentally, Thurston was Weeks' adviser in graduate school. Definitely try to get the second edition, which includes a discussion of Weeks' involvement in ongoing attempts to figure out the actual shape of our universe. Somewhere between the first and second editions of this book, Weeks won a MacArthur grant.

- Euclid, "The Elements" (3 vols.), Dover, 1956.

One of the greatest works of Western Civilization, and one of the most boring as well. It's unquestionably true that our lives would be very different today if this book had not been written, but that's no reason to read the whole thing. It is well worth knowing what Euclid was trying to do, how he did it, and whether or not he succeeded, but it doesn't take all three volumes to get that. Look it over, in any case. This used to be required reading for every person claiming to be educated. Unfortunately, less Euclid in schools has not been replaced by other kinds of geometry.

- Robin Hartshorne, "Geometry: Euclid and Beyond," Springer-Verlag, New York, 2000.

One of the most impressive mathematics textbooks I have recently seen. This text, meant as a companion to Euclid's ``The Elements,'' does not summarize Euclid, but rather explains what his conceptual understanding was and how it differs from our contemporary approach, and shows how Euclid can be brought mathematically up to date. Though most of the book is aimed at an audience of junior or senior level college mathematics majors (and, in particular, makes use of abstract algebra), much of the discussion of Euclidean geometry in the first two chapters is accessible to a broader audience, and well worth the price of having to skip over some technicalities. Hartshorne has done an astonishing job of figuring Euclid out, making this a substantial book with equally substantial rewards.

- Benoit Mandelbrot, "The Fractal Geometry of Nature," W. H. Freeman, 1977.

Something of a modern classic by the man who helped develop fractals into an important subject (though he did not invent it), and who single-handedly brought fractals to the attention of the world. The pictures are great, but like much pioneering work it's rather difficult to figure out what's going on. Give it a try, but don't be surprised if you'll need to look elsewhere to find out what fractals actually are.

- Peter R. Cromwell, "Polyhedra," Cambridge University Press, Cambridge, 1997.

A lovely text for the non-specialist. There is a wealth of historical information on the study of polyhedra, wonderful illustrations, and an excellent choice of topics, ranging from such standards as the Platonic solids to less well known (for a popular audience) gems such as Descartes' Theorem on angle defects and Connelly's flexible sphere. The one real drawback is the lack of exercises for the reader, devaluing this book as a textbook, but well worth reading nonetheless.

- Colin Adams, "The Knot Book," W. H. Freeman, NY, 1994.

The non-mathematician may think of knots as something only sailors and boy scouts care about, but in fact there is a fascinating mathematical theory of knots that plays an important role in the field of topology, and has some applications to such things as DNA and theoretical physics. It is hard to imagine a better place for the non-specialist to start learning about knot theory than this carefully written text. The material is not always easy (given the often technical nature of the subject), but the book avoids as many technicalities as possible, has lots of exercises for the reader scattered throughout, has nice illustrations, and a readable style aimed at helping the reader as much as possible. There are even a few knot theory jokes at the end of the book. Who could ask for more.

- Marjorie Senechal and George Fleck, "Shaping Space," Birkhäuser, Boston, 1988.

This book is the proceedings from a conference on various aspects of polyhedra and related topics, which might sound dull until you take a look at itlooking through this book sure makes me wish that I had been at that conference! Though a few of the article are quite technical, many are aimed at a general audience, including a nice history of the study of polyhedra. The book is very well illustrated. I wouldn't necessarily recommend buying this one unless you are a hard core polyhedra fan, but it is well worth a browse.

- George E. Martin, "Transformation Geometry," Springer Verlag, 1982.

A very technical book appropriate for people with at minimum some Calculus level mathematics (though Calculus per se is not required). The book has a nice treatment of frieze and wallpaper groups, tilings, and projective geometry. This one demands serious study.

- D. Hilbert & S. Cohn-Vossen, "Geometry and the Imagination," Chelsea, New York, 1956.

A genuine classic of mathematical exposition for a broad audience. The first named author was one of the truly great mathematicians of the early 20th century, and it shows in this lovely book that covers a broad spectrum of geometric ideas, ranging from polyhedra to surfaces to linkages to topology. This text is a bit more technical than some others aimed at a general audience, but the reward is commensurably greater for those who put in the effort.

- Edwin A. Abbott, "Flatland," Dover, 1952 (or other editions).

A minor classic, with heavy emphasis on both words. "Flatland" recounts the adventures of A SQUARE, who lives in a 2 dimensional world. The first part of the book, a satire of the Victorian society in which Abbott lived, describes the racist and sexist social order in which our hero lives. The second part of the book describes A SQUARE'S encounter with lower and higher dimensional beings, thus introducing the reader to some important ideas about the fourth dimension and higher. Neither great writing nor brilliant mathematics, "Flatland" straddles the fence so well that its place in the canon is assured. (Be careful with the introductions to various editions of "Flatland" -- the one by Banesh Hoffmann in the Dover edition, and the one by Isaac Asimov in the HarperCollins edition, both entirely miss the point of the book.)

- Dionys Burger, "Sphereland," Perennial Library (Harper & Row), 1965.

A modern sequel to "Flatland," introducing many mathematical ideas recognized as important since the advent of Einstein's theory of relativity (which post-dates "Flatland" by 25 years). "Sphereland" was written by a mathematician, which shows in both the well chosen mathematical topics, and the less than gripping narrative style. Though mathematically more substantial than "Flatland," it lacks the latter's bite.

- Rudy Rucker, "The Fourth Dimension: a Guided Tour of the Higher Universes," Houghton Mifflin, 1984.

A fun book covering a lot of serious material, and some rather esoteric stuff to boot. Rucker makes higher dimensions, relativity and geometry enjoyable and surprising in a way no one else can. Lots of good problems and puzzles, and great quotes and illustrations. Come to your own conclusions about the more speculative stuff -- I'm sure Rudy wouldn't have it any other way.

- Thomas F. Banchoff, "Beyond the Third Dimension," Scientific American Library, NY 1990.

I wish I had written this one, though it is just as well that I didn't, since it is hard to imagine that anyone else would have come close to doing it as well as Banchoff (a serious research mathematician with a genuine interest in reaching a broad audience). This book is such a carefully thought out and beautifully illustrated treatment of higher dimensions that it could make a fine coffee table book, though don't let that fool you -- this book discusses serious stuff. The excellent choice of topics range from unfolding and slicing higher dimensional cubes (with great computer graphics) to perspective and scaling. After reading the classics "Flatland" and "Sphereland," this would be an excellent next place to which to turn if you want to know more about higher dimensions.

- A. K. Dewdney, "The Planiverse," Poseidon Press, 1984.

A very detailed exploration of what a 2-dimensional world would really be like. The emphasis is not on mathematics (as in Flatland), but on physics, biology and technology in 2 dimensions. What would a 2-dimensional sailboat look like? How would 2-dimensional intestines keep from splitting a creature in two? It's all quite fun, though a bit more than you might want to know.

- Michio Kaku, "Hyperspace," Anchor, 1994.

A rhapsody about the latest theories of physics (e.g. string theory, parallel universes, wormholes and the like), and their relation to mathematics, especially the study of higher dimensions. Written by a physicist, it has the advantage of an insider's view of the latest physical theories, and the disadvantage of a physicists view of mathematicswhich to a mathematician seems a bit distorted. The first few chapters on higher dimensions contain some interesting historical discussion of the rise of popular interest in the subject, but the mathematical ideas can be found treated better elsewhere. If you want to learn about physics, then by all means read this book.

- Charles H. Hinton, "Speculations on the Fourth Dimension," Dover, 1980.

Probably a must for hard-core 4th dimension fans, but not necessarily for anyone else. Hinton, a mathematician obsessed with the 4th dimension, wrote a variety of essays and "Flatland" style stories that have been excerpted and collected by Rudy Rucker in this volume. The fiction attempts to be more scientific than "Flatland" (having a different sort of 2 dimensional world, and anticipating the later book "The Planiverse"), but the narrative is tedious, and imbued with Hinton's mystical ideas. The essays are fine in part, but, as with the fiction, there is better elsewhere.

- Leon Battista Alberti, "On Painting." Yale, 1956.

A Renaissance classic. It's a bit hard to follow, but the excellent introduction and notes by the translator help a great deal. The actual part about perspective painting is very short, but lays the groundwork for much to come.

- William M. Ivins, Jr., "Art and Geometry," Dover, 1946.

An interesting little book. Ivins has a point about the ancient Greeks that he tries to make at all costs, and it does get a little tedious by the end; the book is quite thin, however, so don't let that stop you. His views are provocative, and worth considering even if you end up not agreeing. The book is completely nontechnical, though some pictures would have been helpful.

- Martha Boles & Rochelle Newman, "The Golden-Relationship, Book 1," Pythagorean Press, 1987.

A workbook that actually has you get your hands dirty with geometric constructions using straightedge and compass. In between the problems and projects are very readable discussions of the Golden Ratio, Fibonacci numbers and the like. Some readers may find the philosophical exposition a bit flaky, but it's worth wading through it for the sake of the hands-on approach. Besides, how can you go wrong with a book that has a recipe for "Fibonacci Fudge."

- Theodore A. Cook, "The Curves of Life," Dover, 1914.

More than you ever wanted to know about spirals, from rams' horns to spiral staircases. The first and last few chapters are worth reading; the stuff in between (which is a fair bit) makes for fun browsing.

- Matila Ghyka, "The Geometry of Art and Life," Dover, 1977.

In spite of the broad title, most of the book focuses on the Golden Ratio and related topics. Some of the material is good, though a bit technical; other parts of the book are a bit speculative (to put it politely), concerning various esoteric theories the author appears to believe. Interesting reading if you can deal with it.

- H. E. Huntley, "The Divine Proportion," Dover, 1970.

A rhapsody about the Golden Ratio (a.k.a. the Divine Proportion), and beauty in mathematics in general. Some of the material is philosophical, some fairly technical. It's worth picking bits and pieces out of this book.

- Robert Lawlor, "Sacred Geometry," Crossroad, 1982.

Great pictures, and all kinds of esoteric theorieswith lots of geometrical constructions. You will have to decide for yourself what's going on here, since I am not sure.

- Dan Pedoe, "Geometry and The Visual Arts," Dover, 1976.

A nice choice of topics, and good illustrations, but the style consists of much aimless ramblings, and at times it's hard to figure out what the point of it all is.

- Douglas Hofstadter, "Gödel, Escher, Bach," Vintage, New York, 1979.

I once heard this book referred to as the least-read best-seller ever, and that may well be true. The title is appealing, and the book is astoundingly clever (and potentially very rewarding if you slog through it), but this is not the fun and easy read about the intersection of mathematics, music and art that the title might suggest. The focus of the book is mainly Gödel's remarkable -- and technically brilliant -- Incompleteness Theorem in logic, which says that in any mathematical system that contains arithmetic, there will always be theorems that can neither be proved nor disproved. The relation to Bach and Escher is the issue of self-reference, which is crucial to Gödel, but hardly the most important issue for the other two. My recommendation: Spend more time listening to Bach's music than reading about it; find an easier exposition of Gödel's work; and for fun take an occasional peek at Escher (who is not at all in the same league as the other two).

- Wassily Kandinsky, "Point and Line to Plane," Dover, 1979.

I bought the book because of its author (a prominent 20th century artist) and title. I have no idea if this book is of any use to practicing artists, but the only mathematics I could find in it is the appearance of words like "line," "angle," etc. In other words, there is no real mathematics in this book. Maybe there isn't supposed to be, so it's all right.

- Taylor, Alan, "Mathematics and Politics: Strategy, Voting, Power, and Proof," Springer-Verlag, NY, 1995.

Beneath all the hype, money and nastiness of politics is a mathematical theory that plays an important role in such political questions as choosing the most appropriate voting system when there are more than two candidates (the standard American system of whomever gets the most votes wins is not considered the best); apportioning seats in the House of Representatives; and calculating the relative power of members of committees. This is fascinating material, and Taylor (a mathematician) has done an admirable job of making it accessible.

- Robert Geroch, "General Relativity from A to B," U. of Chicago, 1978.

The nicest elementary exposition of General Relativity I have seen (although I haven't seen all that many, being a mathematician rather than a physicist). The explanations are with words and diagrams, rather than formulas. A masterful effort at being simple and understandable without being cute, silly or condescending.

- Georges Ifrah, "From One to Zero: A Universal History of Numbers," Viking, 1985.

Numbers are both useful and fascinating, with an interesting history that is not often taught in school. Unfortunately, this book is not really about the conceptual history of numbers, but rather the history of numeration, that is, the ways in which numbers have been written and calculated with. I find this material rather boring, but if you are interested in it, wonderful, but even then I would be hesitant about this book, even though it looks appealing at first glance -- I have read a rather scathing critique of its historical accuracy, though I am not a historian, and cannot judge for myself.

- Herman Weyl, "Symmetry," Princeton, 1952.

A classic by one of the great mathematicians of the 20th century. The caliber of the philosophical and historical discussions reflect the stature of the author. Weyl does lapse into some overly technical passages, but they are well worth wading through for the rest. Great illustrations as well.

- Farmer, David, "Groups and Symmetry," AMS, 1996

The idea is of this book is great: an exposition of lovely mathematical topics including symmetry, ornamental patterns and groups, aimed at non-mathematicians, done not by lecturing but by brief discussion combined with lots of 'tasks' for the reader to explore. Unfortunately, the writing is at times awkward, the choice of terminology is on occasion unfortunate, the organization is poor and the 'tasks' vary from trivial to extremely hard with no warning. A few extra revisions would have helped. A well meaning book that does not quite live up to its promise.

- Branko Grünbaum & G. C. Shephard, "Tilings and Patterns," W. H. Freeman, NY, 1987.

The ultimate reference on the mathematical theory of tilings and other planar ornamental patterns, this massive book will surely be the definitive source in the foreseeable future. Though most of the text is mathematically sophisticated, the lovely introduction is accessible to all, and the pictures and figures throughout the text are great. I would not recommend buying this one unless you are planning a serious study of the subject, but it is well worth looking through.