Read This!

The MAA Online book review column

Briefly Noted

August 2006

At this point, we are adding some 30 reviews a month to MAA Reviews. This column contains (sometimes abbreviated versions of) some of the reviews that have appeared there (fairly) recently.

In the mid-1970s, Martin Gardner and Robert Tappay came up with the idea of producing filmstrips for use in high school classrooms, one about mathematical paradoxes and one about mathematical problems that can be solved easily provided one has just the right insight. The strips featured illustrations by Jim Glen; they were marketed as "The Paradox Box" and "The Aha! Box".

Of course, what Gardner and Tappay didn't realize (hardly anyone did, at the time) was the filmstrip was already dead, about to be replaced by fancier technologies such as video. (Some readers of this review may not even know what filmstrips were…) So the two "boxes" were transformed into books, called Aha! Gotcha and Aha! Insight, originally published in 1982 and 1978, respectively, which are reprinted here in one volume. Both books follow basically the same format: the illustrations from the strip are reproduced together with the text that was supposed to go with them, and then commentary is added, either on the side of the page or on a facing page. The images, unfortunately, are a little small, but the effect is still quite interesting.

I remember Aha! Gotcha from way back when, particularly for two limericks. The first went

There was a young lady of Crewe
Whose limericks stopped at line two.

and the second was

There was a young man of Verdun.

I always found that last one delightful. (How many lines does it actually have?)

Gotcha starts with paradoxes from logic and set theory, then goes on to cover numbers, geometry, probability, statistics, and time. Insight classifies its puzzles along similar lines: combinatorial, geometric, numerical, logical, algorithmic, and lexical.

Most mathematicians and mathematics fans are familiar with most of the paradoxes in Gotcha , and many of the problems in Insight are also well known, particularly to readers of Gardner's other books. On the other hand, since this way of presenting them was originally intended for class use, and in high school to boot, the book contains a lot of nice elementary ways to introduce some interesting mathematical ideas. Some of these pages could be made into really neat little PowerPoint presentations. (Of course, if you plan to reproduce the images, you need to get permission from MAA!) If you ever use puzzles or paradoxes in elementary classes, you can find some help here.

Martin Gardner is a national treasure, someone whose contribution to mathematics has been immense. The books collected here, while not his best, are accessible, intelligent, and fun. If you didn't happen to buy them in 1978 and 1982, here's your chance.

[Fernando Gouvêa; posted to MAA Reviews 07/31/2006]

Beautiful Evidence, the latest book by Edward Tufte, lives up to its name. The book itself is strikingly beautiful, starting with the dust jacket, continuing through the page design and layout and, most importantly, the many reproductions of excellent, mediocre and abysmal examples of graphical presentation of data. Indeed, this book would be worth its purchase price merely for the illustrations, which Tufte informs us "come from 14 centuries, 16 countries (Italy and France, especially), 3 planets, and the innumerable stars." The choice of such varied illustrative manner is no accident: Tufte states in the introduction that he believes that "the principles of analytical design are universal" and this book is sort of an extended discourse on that theme. Tufte believes that "making an evidence presentation is a moral act as well as an intellectual activity" which explains in part why he reacts so strongly to poor or misleading graphical presentations.

Edward Tufte is most noted today for his work in the field of informational graphics. His previous books include The Visual Display of Quantitative Information (1983; 2nd ed. 2001), Envisioning Information (1990), and Visual Explanations (1997). Further information, including a moderated forum and a number of articles by and about Tufte, are available on his website: http://www.edwardtufte.com.

But it is difficult to imagine that anyone with a serious interest in the graphical presentation of information would not already be familiar with Tufte's work. If they do exist, such readers may become confused by what sometimes appears to be the random presentation of topics, and by the far-reaching generalizations of the author. Beautiful Evidence is a very individual book, and anyone who buys it expecting a standard textbook on graphic design will be disappointed.

The apparent disorganization is deliberate: one of the evils Tufte is fighting is a mechanical approach toward data presentation, which encouraged by cookbook methods presented in many textbooks and by many software programs commonly used to produce statistical graphics. Rest assured, there are in fact unifying principles behind the many examples, although the connections between the various examples may not always be immediately obvious. Readers who want an overview of Tufte's principles can start by reading the chapter "The Fundamental Principles of Analytical Design." Otherwise, I recommend starting anywhere and stopping to carefully study those examples which are most personally appealing. For instance, I particularly enjoy Tufte's suggested revisions of published graphics, such as those from Markus Bloch's Ichthyologie and from Carl Sagan's The Dragons of Eden.

Tufte does have a tendency to express himself in extreme terms, but taking issue with particular statements or principles seems to me to be beside the point. We can argue forever about whether the principles of analytical design are really universal, but it is not necessary to settle that issue in order to benefit from the material presented in Beautiful Evidence. The occasionally harsh tone of Tufte's criticism is offset by his considerable wit.

Readers who have read Tufte's previous books and articles on graphic design will find that some of this material is familiar, particularly the chapter on "The Fundamental Principles of Analytical Design" which is based on Charles Joseph Minard's data-map of the manpower losses of the French army during their 1812 invasion of France, and the chapter "The Cognitive Style of PowerPoint: Pitching Out Corrupts Within," much of which was previously published in pamphlet form.

[Sarah Boslaugh; posted to MAA Reviews 07/24/2006]

Project Origami, by Thomas Hull, is quite an interesting idea. Hull, a mathematician, uses origami to gain insight into various branches of mathematics.

The book is written in the style of a primary school mathematics workbook. It's divided into twenty-two "activities," each of which is subtitled with the mathematical topics it covers. The span of topics is astounding, ranging from the obvious, such as Geometry, to the unexpected, such as Number Theory. There's even the amusing "math for liberal arts" subtitle.

It's hard to pin down the intended audience. While folding paper may seem silly to some college students, the math behind the folds can actually be pretty sophisticated. Some of the activities, such as "Folding a Parabola" could be used to assist precalculus students to visualize and understand quadratic equations. Some of the more difficult activities might require more mathematical sophistication for a full understanding, but might also be wonderful way to give younger students a glimpse of mathematics. Conversely, some of the topics go so far as to call for the use of abstract algebra for a fully general explanation.

Overall, this book is an excellent resource for mathematics educators who would like to include some hands-on experimentation in their teaching. While its most obvious use is to present some interesting topics of mathematics to calculus-age students, it contains enough mathematical sophistication for most undergraduates, provided that they can take it seriously enough.

[Steven Frankel; posted to MAA Reviews 07/27/2006]

Publication Data

Beautiful Evidence, by Edward R. Tufte. Graphics Press, 2006. Hardcover, 213 pages, $52.00. ISBN 0961392177.

Project Origami: Activities for Exploring Mathematics, by Thomas Hull. A. K. Peters, 2006. Paperback, 245 pages, $30.00. ISBN 1568812582.

Fernando Q. Gouvêa remembers the Aha! books from the 1980s, but he lost them along the way, so he's glad to have them back.

Sarah Boslaugh is a Senior Statistical Data Analyst in the Department of Pediatrics at the Washington University School of Medicine in St. Louis, MO. She wrote An Intermediate Guide to SPSS Programming: Using Syntax for Data Management with Sage Publications in 2005 and is currently writing Secondary Data Sources for Public Health: A Practical Guide for Cambridge University Press. She is also Editor-in-Chief of The Encyclopedia of Epidemiology which will be published by Sage in 2007.

Steven Frankel is an undergraduate Engineering student at The Cooper Union in New York, NY. His primary interests lie on the line between Electrical Engineering and Mathematics, including Signal Processing and Control Systems. He can be contacted at franke2@cooper.edu.

Back Issues:

• April 1999: Reprints of "classics", books on applied mathematics.
• May 1999: Books on math and physics, both popular and technical.
• July 1999: Various kinds of geometry, and classics in applied mathematics.
• August 1999: What's happening, Dirichlet, and Cryptonomicon.
• October 1999: Graph theory sources, John von Neumann, and Noncommutative Geometry.
• November 1999: Number theory in nine chapters, mathematics at work, and Jacques Hadamard.
• February 2000: Fallacies, Flaws, and Flimflam, measuring the universe, and primary sources at USMA.
• March 2000: Cardano and Manifold: Time.
• May 2000: Grassmann, Smarandache, and a really big prime.
• June 2000: A problem book, a history book, a collection of biographies, and a historical survey of reciprocity laws.
• August 2000: acrostics, field theory, categories, and partial differential equations.
• September 2000: excursions, history, and some high-level expository writing.
• November 2000: a festival of math videos and the Fourier-analytic proof of quadratic reciprocity.
• December 2000: Analysis problems, mathematical mysteries, and industrial mathematics.
• January 2001: Euler, Kolmogorov, and Honsberger.
• March 2001: history of mathematics, in three flavors: Harriot, prime numbers, and geometrical exactness.
• May 2001: three very different "applied mathematics" books.
• August 2001: introductions to algebraic number theory and to topology, and problem books.
• November 2001: Wilbur Knorr, John H. Conway, and vector analysis.
• February 2002: biological time, celestial mechanics, and things rarely talked about.
• March 2002: history of recent mathematics and differential geometry.
• June 2002: mathematics in other cultures, Archimedes, probabilistic thinking, and Dover's Phoenix.
• August 2002: Euclid's Elements, Hurewicz's Lectures, and Olympiads 2001.
• October 2002: Algebra from A to Z, Problems in Probability, and a really old trigonometry textbook.
• December, 2002: group representation theory, Irish mathematicians, and integration.
• January, 2003: careers, logic, dueling idiots, and differential geometry.
• March, 2003: the Putnam, the Liber Abaci, and the Millennial Conference.
• April, 2003: analysis, reprints, and Rithmomachia.
• May, 2003: conversation starters, dissections, and Fibonacci numbers.
• July, 2003: mathematical circles, the Scottish Café, and the zeta function.
• September, 2003: history — an encyclopedia, historiography, and algebra.
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• December, 2003: calculus, relativity, and biology.
• February, 2004: cryptography, generating functions, Wallis, and Stirling.
• March, 2004: differential equations, fundamental groups, reprints, and spherical buildings.
• April, 2004: Ramanujan graphs, time, and Medieval Islam.
• May, 2004: History: of Greek mathematics, of mechanics, of tables, of Reverend Bayes.
• June, 2004: The Grothendieck-Serre correspondence, mathematical biology, olympiad problem books, and units.
• July, 2004: Groups, Probability, Olympiads, and Isaac Newton.
• October, 2004: Klein, dogs, rivers, physics, geometry.
• November, 2004: Karl Pearson, chessboard problems, John Wallis, and contemporary art.
• January, 2005: Abel's legacy, reductive groups, curriculum foundations, and irresistible integrals.
• February, 2005: Differential equations, graphs, optimization, algorithms... and arithmetic on function fields of characteristic p.
• March, 2005: William Rowan Hamilton, π, Adolph Zeising, and Jesuits.
• April, 2005: differential equations, abstract algebra, and two kinds of number theory.
• May, 2005: algebra, chance, variations, and Chinese.
• Late May, 2005: motives, historical sources, Lebesgue, Cauchy.
• July, 2005: calculus, control theory, instabilities, biostatistics.
• August, 2005: difference equations, Galois theory, algebra, martingales in finance.
• October, 2005: 4-manifolds, geometry, Turing, and statistics in sports.
• December, 2005: lattices, irrational numbers, topology, and linear models.
• January, 2006: algebra, geometry, bubbles, jaws, Greek mathematics, and quotations.
• February, 2006: classics in statistics and in mathematics education, a math league, and transcendence.
• March, 2006: Halmos, Riemann, statistical models, and a contest problem book.
• April, 2006: Real analysis, maxima and minima, geometry, and the IMO.
• May, 2006: numbers, statistics, algebraic numbers, and problems from New Mexico.
• June, 2006: pure math, technical shop math, and étale coverings.
• July, 2006: probability, geometry, cryptography, and Huygens.

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Read This! is the MAA Online book review column. Contributions are welcome; contact the editor if you'd like to be one of our reviewers. Books for review should be sent to the editor: Fernando Gouv&ecric;a, Dept. of Mathematics, Colby College, Waterville, ME 04901. Publishers, please check our reviews information page.