Perspective in Math and Art

As a mathematician, I expect that people at parties will tell me that they're no good at math. I'm used to my fellow professors confessing their ignorance of my subject. I understand that many of my students think math is hard and scary. That's why I was so eager to do drawing -- something I figured would be easy and approachable -- in my math classes.

But to my great surprise, I found that it is the art, not the math, that makes people nervous. As my co-author Marc Frantz told me, most college graduates have a bit of math in college, and almost all have had a math class their senior year of high school. But few adults have had an art class since 6th grade. Carra’s drawing below is typical of what I see at the beginning of the semester in my course. My students enter college drawing like children, and they are understandably embarrassed by this.

Carra's first drawing

I got started with a math-and-art project because of a fortunate coincidence of time and space: I happened to be spending part of my first research leave at Indiana University-Purdue University at Indianapolis during the semester that Indiana University was applying for a large Mathemematics Throughout the Curriculum grant from the National Science Foundation. The MATC project aimed to team up a mathematician with an "other" — a historian, a biologist, an economist — to do math in the context of that other discipline. Marc Frantz was in the math department at IUPUI then (in fact, he was the executive director of the MATC project), but he had his first graduate degree from the the university’s Herron School of Fine Art. We teamed up; I added "instant dissemination" by taking the project out of Indiana and back to Franklin & Marshall College, my home institution. I started designing a first-year seminar, a course that our college uses to introduce students to writing and introductory research skills, but that leaves me a lot of flexibility with regard to content.

When I started working with Marc on designing a course on the mathematics of art, I didn't realize Marc would soon have me looking at the world in a whole new way — literally. Here you see Marc's students looking through a window at buildings outside, directing their classmates to recreate the image of those buildings on the windows themselves. (Drafting tape is easily removable, for which the custodial crews thank us!)

I'm just a math geek, but over the past decade while we were writing Viewpoints: Mathematical Perspective and Fractal Geometry in Art and leading workshops for other instructors, Marc and I have gotten to repeat the window-taping exercise with an amazing list of 200 people. I've taped windows with mathematicians and artists, with chemists and geologists, with a minister and a motorcycle rider. One couple who came to our Pennsylvania-based workshop stuffed their dorm room here with shrubbery they'd take back to Ohio at the end of the week. Other instructors taught my student helpers to play an electronic party game called "Catch Phrase," and it went viral that week.

The most enjoyable part of this project, though, has been seeing my students wrestle with simple-seeming questions (where do we draw the next fencepost?) and come up with those Ah-HA! moments of insight. In our book you'll see statements and theorems listed by number, but my students and I think of them as "Alex's Theorem" and "Dierdre's construction." We all ought to get a chance to name a brilliant insight after ourselves or our friends, I think.

Gary Larson, in one of his Far Side comics titled "Math Phobic’s Nightmare," shows Saint Peter quizzing a supplicant at the Pearly Gates with this question: "O.K., now listen up. Nobody gets in here without answering the following question: A train leaves Philadelphia at 1:00 p.m. It’s traveling at 65 miles per hour. Another train leaves Denver at 4:00.... Say, you need some paper?"

Larson’s nightmare is in perfect contrast to why our work with students has been so much fun. We all know and can parody the dreaded two-trains problems. A simpler question is this: If you sketch a picture of the rails of the train track going into the distance, and you know where the first two railroad ties go, where do you put the next one? In our class, we change the problem from a horizontal one to a vertical one: the question becomes "Where does the next fence post go?"

Where does the next fencepost go? (Hint: not at the point marked ‘P ’).

Where does the next fencepost go? (Hint: not at the point marked P).

It’s an easy question to understand, and that simplicity itself makes it unusual in mathematics. It’s an obvious question: any artist would want to know the answer. It’s even a question that begs to be answered – few people care at all when those two trains meet, but if we want to draw a decent-looking picture of a sidewalk or a fence, then this question about where the next line goes is going to matter to us. It’s not obvious what the answer is ... and that puzzle of obscurity lies at the heart of mathematics. This problem is puzzle-solving at its prettiest.

There are a lot of other aspects of this puzzle (and of related drawing puzzles) that I have come to love. One is that artists often answer the puzzle long before their mathematical counterparts – this trend holds at the undergraduate level all the way up through Ph.D.-holding professors. We mathematicians tend to stare at the paper, hoping an answer jumps off the page at us. Artists pull out their pencils and start doodling, often stumbling upon a solution almost by accident. The artists have learned to overcome their fear of drawing something that is "wrong," and so they become the first ones to draw a solution that is right.

Another aspect of this problem I love is that, although there is only one correct answer, there are many different correct ways of arriving at that answer. In this way, the problem captures the essence of mathematical research. The bane of my profession is the student who likes math for the black-and-white-ness of the subject: "because there is always one right answer." With this fence problem, however, I can celebrate my students’ various solutions, exploring nuance and expounding upon elegance.

It's also a lovely problem because, once you know a way to sketch the answer, it’s fun to do over and over ... drawing the fencepost lines going into the distance becomes almost like a meditation. (Not many people would say the same about bringing together those trains from Philadelphia and Denver).

But Viewpoints hasn’t just helped me and my students draw the world around us; it’s helped us look at the world around us. When a person tapes an outline of what she sees onto a window, the only way another person can make the taped picture line up with the outside world is to stand exactly where that artist stood. In the same way, we can reconstruct where long-ago artists must have placed themselves in their paintings. By standing in the same spot in front of the canvas, we see the paintings become full and more 3-d than ever. We do this without 3-d glasses, without focusing or unfocusing our eyes near a stereogram. Really, the only "trick" is to understand mathematical proportions.

When I learned this trick of finding the right viewing location, I finally understood why all my vacation pictures failed to capture what I thought I saw when I took the pictures. My scenic photographs would come back looking technically correct but empty of that majesty that surrounded me when I gazed on that canyon or that field. The problem isn’t in my camera or my technique. It is a problem of simple proportions: the lens was close to the film but I’m not close to my photograph. I can "fix" this problem in one of two easy ways, either by enlarging the photographs, or by moving my eye in very close to the small photo.

You can try this yourself with this simple image of a box from our book. It looks more like a brick or a dumpster than a cube to you, I’m guessing. But if you enlarge the picture a lot on your computer, or if you lean in very, very close to that top right tree and look down at the box, you’ll see the proportions appear to change, and this brick will become very cube-like. It’s mathematical magic.

Is this a brick, or is this a cube? What you see depends on how close you are to this picture.

Is this a brick, or is this a cube? What you see depends on how close you are to this picture.

Learning the mathematical "rules" for drawing opens up whole new possibilities. In this context, rules don't stifle creativity; they allow for fuller expression. My math-and-art students have flourished, and I have been heartened, too. Few of my students ever want to see their final calculus exam after they turn it in, but almost all of my students show their parents photocopies they've made of the final drawings they've turned in to me. Carra’s final drawing, like so much of my students' late-semester work, shows a mastery of space with hints of great things beyond the horizon. You can tell she's not going to be afraid of anything.

Annalisa Crannell
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Annalisa Crannell is a professor of mathematics at Franklin & Marshall College. She and Marc Frantz, a research associate at Indiana University-Purdue University at Indianapolis, are co-authors of Viewpoints: Mathematical Perspective and Fractal Geometry in Art, recently published by Princeton University Press.

Writing About Math

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At MIT, professors grapple with how to teach students to communicate about numbers in clear English.

The New Math

As a kid, my favorite book in the world was E.T. Bell’s Men of Mathematics (1937). I must have read it dozens of times by the age of 14. One afternoon, coming home from the library, I could not resist opening the book to a particularly interesting chapter -- and so ended up walking into a parked bus.

With hindsight, certain problems with the book are clear. Bell’s approach to the history of mathematics was exciting, but he achieved that effect, in part, through fictionalization. We now know that embroidering the truth came as second nature to Bell, who was a professor of mathematics at the California Institute of Technology until shortly before his death in 1960. In addition to writing science fiction under a pseudonym, Bell also exercised a certain amount of creativity in telling his own life story – as his biographer, Constance Reid, found out through some detective work.

But another problem with Men of Mathematics only dawned on me recently. I hadn’t thought of the book in ages, but remembered it while reading while reading Letters to a Young Mathematician by Ian Stewart, to be published next month by Basic Books.

The author is a professor of mathematics at the University of Warwick in the U.K. The imaginary recipient of his letters is named Meg -- a nice departure from the longstanding and rather self-reinforcing stereotype of math as a man’s field. The idea that no gender has a monopoly on mathematical talent seems never to have occurred to E.T. Bell. (Nor, consequently, did it cross the mind of a certain young nerd colliding with stalled vehicles during the mid-1970s.)

Fortunately that situation has started to change. And the progress is reflected, in a quiet and matter-of-fact way, in Stewart’s Letters.  

A story unfolds, chapter by chapter, as Stewart corresponds with Meg. In the earliest letters, she is still in high school. By the end of the book, she has tenure. It is, in effect, a bildungsroman at one remove. The reader watches over Stewart’s shoulder as the young mathematician turns an early talent into a stable professional identity.

There’s even a moment when, in search of an interesting project to test her abilities, Meg starts trying to find a method for trisecting an angle using only a compass and an unmarked straightedge. This is one of the problems handed down from ancient geometry. People “discover” solutions to this challenge all the time, then become indignant that mathematicians don’t take them seriously. (The proof of why it is impossible involves mathematical tools going way beyond anything available in antiquity.)

But most of the guidance Stewart offers is positive -- and some of it seems useful even for those of us without mathematical aspirations or gifts.

“My usual method for reading a mathematics text,” he recalls about his student days, “was to thumb through it until I spotted something interesting, then work backward until I had tracked down everything I needed to read the interesting bit. I don’t really recommend this to everyone, but it does show that there are alternatives to starting at page 1 and continuing in sequence until you reach page 250.”

The most surprising thing -- at least for anyone influenced by Bell’s romanticized account of the mathematical vocation -- is Stewart’s emphasis on the nuts and bolts of academic life. Letters is full of pointers on academic politics, the benefits and frustrations of collaboration, and how to avoid disaster at conferences. (“Never believe your hosts when they tell you that all the equipment will work perfectly,” he notes. “Always try it yourself before the lecture.”)

E. T. Bell told stories about mathematicians whose lives were shaped, in the final analysis, only by their own creative instincts. They might occasionally win a prize offered by a learned society, or feel driven to some breakthrough by the challenge of defeating a hated rival. But Bell’s men of mathematics were, on the whole, geniuses of the purest vintage. They had inspirations, not resumes. It is hard to imagine anyone trying to give Carl Friedrich Gauss useful career advice.

So does that mean that popularized accounts like Bell’s are something a young mathematician ought to avoid? I contacted Stewart by e-mail to ask his thoughts on the matter.

“I write a lot of books popularising math and science, so I may be biased,” he said in reply, “but when I was in high school I read all the books I could find about the history of math, about mathematicians, and about various topics in math. And those definitely had a significant effect on my interest in the subject. They made it clear that math has a long and fascinating history, that the great mathematicians were real people, not just obsessed geniuses who couldn't tie their own shoelaces, and that there is much, much more to math than the tiny part of the subject that we are all taught at school.”

Well, that’s a relief. There’s something to be said for idealization and hero worship, after all, in their proper season. You then have your whole life to become more realistic, not to say more calculating.

Scott McLemee
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Let the Restoration Begin

President Obama promised in his inaugural address to “restore science to its rightful place” and “transform our schools and colleges and universities to meet the demands of a new age.” These were refreshing and uplifting words from a president after the long and dark night to which science and its findings had been relegated during the previous eight years.

But these words also represent no small task to the science-friendly president. A civil crisis of science illiteracy exists today in America, and Obama’s administration is now charged with undoing a generation of decline in science policy and education in the U.S.

With White House attacks on science behind us for now, science educators must take this opportunity to propose a number of specific goals to ensure and strengthen the politically unbiased use of science in education and policy making.

October 4, 1957, may not be a date that is important to most college students today, but what occurred on this day stunned many Americans at the time. When the Soviet Union launched Sputnik into orbit, all Americans immediately knew that the Soviet Union had silently crept ahead of us in the race to control space.

The American reaction to the 1957 Sputnik launch was much more than rhetoric. The following year Congress tripled the National Science Foundation (NSF) budget to $135 million, and over the next few years of the space race, NSF support reached $500 million. Congress also passed in 1958 the National Defense Education Act, providing funding and scholarships for students and educators interested in science and mathematics.

Not everyone was on board with the new scientific policies, however. During the Kennedy and Johnson administrations industries began mobilizing to defend themselves against new science-based regulations on chemicals, pollution and industrial safety, which threatened to impose large costs on them. After the 1973 Roe v. Wade decision, Christian conservatives also began to mobilize politically. Under Reagan’s leadership, the anti-intellectual, organized efforts to weaken science-based regulations and education only grew stronger.

Then, in the climate of the early 1990s Republican Congress, the Intelligent Design movement grew and flourished, acting through local and state school boards from Kansas to Dover, Pa., to undermine the teaching of evolution. With this foundation, President George W. Bush was able to declare his belief that “both sides should be given equal time” in high school science education.

We have thus seen over the last generation a disheartening trend in science education at the nucleus of our great scientific advancements. In today's education system we import our premier science students from countries like China, India and South Korea. Our secondary school students do not seem adequately trained or even interested in pursuing a rigorous undergraduate curriculum in science, engineering or mathematics. The brightest minds tend to pursue business, law or medicine.

In his inaugural speech, Obama reminded us of the rich and productive relationship between science and public policy that shaped both science education and policy in earlier generations. So far, he has supported his promises with the appointments of distinguished scientists to high-level positions in his administration and by his declaration to reverse the previous administration’s ban on federal funding of research on embryonic stem-cell lines.

Of course, restoring science to its rightful place in government will require more than promises and appointments; it will require sustained hard work.

The conservative coalition will continue to press the same anti-science agenda, constantly seeking lines of attack. And without inherently unbiased infrastructure in the use of science policy, any progress made by the Obama administration can be overturned as quickly as an executive order when next the political tides switch.

Our current crises are no less threatening than the launch of Sputnik was in 1957. Just as investment in science education and research a half-century ago met the Soviet challenge in the Cold War, so, too, can restoration of science education and research as a policy priority help us to meet the demands for cleaner energy, better health and technologically agile national defense on which our future depends.

We thus recommend some specific goals for the new administration, to strengthen the structural support for unbiased use of science in education and policy making:

  • Re-establish the nonpartisan Congressional Office of Technology Assessment, to evaluate science-based policy alternatives.
  • Provide educational institutions a generous budget from Congress to create attractive opportunities for our educators and aspiring students entering the science and engineering curriculum.
  • Renew the federal investment in science education to the level of the post-Sputnik years.
  • Ensure standards in K-12 science education in all 50 states to ensure the teaching of a fact-based curriculum without theistic considerations as central to modern biology.
  • Experiment with new solutions to chronic problems in our secondary schools, to invest in our next generation of young scientists.
  • Restore the importance of good science in the policy setting.

To safeguard the role of science in policy making, the next generation of citizens and science teachers must understand that absolute consensus rarely occurs in science and is not necessary as a basis for policy making. Only a science-literate public can see through such Orwellian discourse as the “junk science versus sound science” false dichotomy. Moreover, science education will help prepare the public for the inevitable controversies that will arise with future scientific advances, as new knowledge sometimes takes us to places where some of us do not wish to go.

The promise of embryonic stem-cell research to cure disease or, more controversially, create desirable physical characteristics, and the search for an energy future freer of carbon, with the uncertain economic implications that entails, attest to the continuing power of science to thrust new issues onto our policy agenda.

The new leadership can and must define science’s role in developing and implementing public policy, and students at all levels of education must be provided with incentives and encouraged to study science to meet, in the president’s words, “the demands of a new age.” They must learn that decision making must be analytical and fact-based in policy-making and that the consequent choices we make remain with us as part of a sometimes messy, always fascinating political process. Let the restoration of U.S. science policy and education begin so that scientific research may be again considered, as it was in our country a half-century ago, the most noble and fruitful of human activities.

Joseph Karlesky and Richard Pepino
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Joseph Karlesky is Kunkel Professor of Government, Richard Pepino is director of the Public Policy Program, and James Strick is associate professor of Earth and environment, all at Franklin & Marshall College.

Beyond the Fields We Know

The most casual of polymaths -- and the most genial (yet precise) of popularizers -- Martin Gardner wrote about science, philosophy, mathematics, literature, magic, and much else besides. He died last month at the age of 95. It is hard to imagine anyone moving into the unique niche he carved out. At the same time, otherwise widely informed people often prove never to have heard of him. I find that even sadder to think now that he is gone.

Readers often discovered Gardner through the monthly column on recreational mathematics he wrote for Scientific American between 1956 through 1981. Numerous books were compiled from it, and they have given stimulation and entertainment to generations of young math nerds. Two years ago, Cambridge University Press began issuing its New Martin Gardner Mathematical Library, with revised and updated versions of many of his pieces. The fourth of its projected six volumes appeared this month.

Gardner only reprinted one of those columns in The Night is Large: Collected Essays, 1938-1995 (St. Martin’s 1996) -- in my opinion, the best book for making an acquaintance with the full range of his interests and talents. There are pieces on theoretical physics, Shakespeare, artificial intelligence, spiritualism, 20th century philosophy, and the language spoken by the Klingons. Gardner published a number of volumes of miscellaneous pieces, all of them enjoyable enough to read, but Night is the best place to start.

Revisiting it a few days ago, I noticed something that escaped me on first reading. Some of the essays originally appeared in The American Journal of Physics, The Journal of Philosophy, and Semiotica, while others were written for newspapers or popular magazines. Yet there is not much difference between them. Proust once said that snobbery never changes its tone even when it changes the subject. The same might be said of Gardner -- with exactly the opposite implication, of course. The tone is generous but precise. He is out to make a point, not to make an impression.

Gardner had originally expected to become a physicist but ended up studying philosophy with Bertrand Russell and Rudolph Carnap as an undergraduate at the University of Chicago during the 1940s. He edited one of Carnap’s books on the philosophy of science and cited him often in his own work. But the sheer range of his interests makes Gardner’s work seem closer to that of Russell's popular writings. They also share certain qualities in their prose – a blend of clarity, familiarity, and humor.

Russell points out in one of his essays that crackpots are an inescapable fact of intellectual life, so you might as well figure out how to derive some entertainment from them. I do not know if Gardner was directly inspired by that insight, but Fads and Fallacies in the Name of Science -- his first book, published in 1952 -- is one of the most diverting volumes I know. New forms of pseudoscience keep springing up, but I suspect Fads and Fallacies will remain in print for a long time to come.

It was where I first learned of -- among other things -- the amazing career and even more amazing discoveries of the great Alfred W. Lawson, who was for many years a minor-league baseball player before turning his attention to the aviation industry, in which he became one of the pioneering entrepreneurs. He had some plausible claim to having invented the first airliner, capable of seating 16 people.

During the Great Depression, Lawson came up with a system of financial reform designed to bring lasting prosperity to his country and the world. Thousands of Americans joined the movement he founded. Gardner located one book from 1941 containing "several hundred photographs of mass meetings, parades, lecture halls, office fronts, bands, and groups of [Lawsonite] officers wearing a special white uniform and cap, and a diagonal sash."

Economics was not his only area of expertise. Lawson also formulated theories about physics and biology, which I must forebear trying to describe in any detail. (Evidently there is no such thing as “energy.” Let’s just leave it at that.) His tireless efforts yielded a comprehensive system of human knowledge called, of course, Lawsonomy.

“In 1942,” wrote Gardner in Fads and Fallacies, “Lawson purchased the University of Des Moines. The school, which included fourteen acres, six buildings, and dormitories for about four hundred students, had been closed since 1929. It is now called the Des Moines University of Lawsonomy... Only Lawson’s own writings are used as texts, and they must be read by a student before he is able to attend. A basketball rule book was once banned because Lawson hadn’t written it. Accredited teachers of Lawsonomy are called ‘Knowledgians,’ and the top-level Knowledgians are Generals. Lawson is supreme head and First Knowledgian.”

Lawson lived for a couple of years after Fads and Fallacies appeared. The Des Moines University of Lawsonomy, which had a peak enrollment of about one hundred students, closed its doors in 1954. It was eventually replaced by a shopping mall. (Another belief system Gardner described in his book, Dianetics, has enjoyed somewhat greater success.)

In 1991, the University of Iowa Press published Zig-Zag and Swirl: Alfred W. Lawson’s Quest for Greatness, by Lyell D. Henry, Jr., a professor emeritus of political science at Mount Mercy College. Its jacket bears an endorsement from Martin Gardner, who called it “one of the most amusing biographies of the last few decades.”

From it you learn that advanced study in Lawsonomy was expected, by its founder, to take about 30 years. It required a narrowness of focus, and a strictness of recall unimaginable, in today's anything-for-a-quick-thrill academe. For one thing, the student had to memorize the great man’s writings. And there were quite a few of them. That is why you don’t run into many really qualified Knowledgians these days. And yet it seems, from YouTube, that there are a few Lawsonians still around. They have a university in Wisconsin. In 2002, enough of its alumni were on hand during a reunion to form baseball teams.

Last year, the University of Iowa Press brought out a paperback edition of Zig-Zag and Swirl. Aside from being the definitive and perhaps final word on the subject, it seems like the most Martin Gardner-esque book ever written by anyone other than Martin Gardner. I got in touch with Lyell Henry to ask for his thoughts about the late author.

“I’ve been a big fan of Martin Gardner’s writing,” he responded, “since 1952, the year in which Gardner’s In the Name of Science was first published (in 1957, the title was changed to Fads and Fallacies in the Name of Science). I had just graduated from high school and by chance found the book in the city library of my home town of Ames, Iowa. Captivated by the sheer zaniness of Gardner’s material, I was especially delighted to find there a whole chapter on Alfred Lawson and his so‑called University of Lawsonomy, the strange institution that I had seen on numerous trips to Des Moines and that had always intrigued and mystified me.”

Henry says he kept up with Gardner’s books and columns over the years. A rereading of the 1952 volume reignited his interest in Lawsonomy. He plunged into research and wrote a book of his own. When the publisher sent his manuscript to Gardner for comment, Henry says he was “greatly relieved to learn that he liked it.”

He considers himself in Gardner’s debt, not just for the inspiration and endorsement he provided, “but, above all else, for providing a superb model of excellent writing that joined logical analysis, clear explication of a wide range of abstruse scientific matters, and, not least, much good humor. This last ingredient -- good humor -- is especially important and one of Gardner’s great strengths. For the past forty years, I have had hanging on the wall of my study a quotation by H.L. Mencken, another writer who knew something about the uses of humor in writing: ‘One horselaugh is worth ten thousand syllogisms.’ I was delighted and gratified to learn not long ago that this was also one of Gardner’s favorite epigrams.”

Scott McLemee
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Graduate Science Enrollments Grow

Smart Title: 
New data show increases for total pool, women, and minority students, but drop in first-time foreign students.

Basic Research and National Security

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A new report warns that declining funds for physics and engineering puts more than the U.S.'s economic status at risk.

Upturn for Science Doctorates

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American universities award more Ph.D.'s, over all and in technical fields, for second year in a row.


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