A Short Primer on Notation in Engineering Dynamics: A Comprehensive Introduction

The authors of the new textbook Engineering Dynamics: A Comprehensive Introduction have written a short Primer to the notation used in their book. The notation differs from that used in the traditional suite of introductory texts (Meriam, Bedford & Fowler, Hibbeler, Beer & Johnson), but this more sophisticated notation is necessary because, as Kasdin explains, this textbook is more comprehensive than anything else currently available. Much of the material that requires this notation, such as multiple frames or three-dimensional rigid body rotation, are not covered in other textbooks.

Kasdin notes that the goal of the primer is threefold: “to show that the notation serves a specific purpose and has pedagogical value, to show that it is not as extensive and different as people think (i.e., it can be categorized into a small number of elements), and that we did not invent it, but rather followed common practice, adopting it from many sources, and merely tried to make it more consistent, systematic, and clear.”

He also emphasizes that there are other more complete and advanced books that have adopted some variation of the notation used in Engineering Dynamics, so there are precedents already in place for their decision to utilize this notation.

You can download the primer here.

Why People Draw features Viewpoints authors Annalisa Crannell and Marc Frantz

Over at Whacky Shorts Creations, they speak with the authors of Viewpoints about their earlier memories of drawing and their current work at the intersection of mathematics, drawing, and art:

Today, I’m so, so excited to present to you a new “why people draw” that is such a wonderful example of how drawing is not just art, but is rather a wonderful visualizing, knowledge-sharing, enlightening thinking tool. Mathematicians Annalisa Crannell from Franklin & Marshall College in Lancaster, PA, and Marc Frantz from Indiana University in Bloomington, IN share their thoughts on drawing, and how they have designed ways to teach math concepts to teachers and college students through drawing! They also discuss how drawing plays a part in their own process of solving problems.

A Biologist’s Guide to Mathematical Modeling in Ecology and Evolution

As mentioned in our earlier post, Sally Otto (new MacArthur Fellowship recipient, yay!) is co-author with Troy Day of A Biologist’s Guide to Mathematical Modeling in Ecology and Evolution. This book is intended for undergraduate biology courses and starts at an elementary level of mathematical modeling, assuming that the reader has had high school mathematics and first-year calculus. It then gradually builds in depth and complexity, from classic models in ecology and evolution to more intricate class-structured and probabilistic models.

Sample chapters of this book are available on our web site:

Chapter 1, Mathematical Modeling in Biology (PDF)
Chapter 6, General Solutions and Transformations—One-Variable Models (PDF)
Chapter 13, Probabilistic Models (PDF)

Supplemental material for instructors is also available online at the authors’ web site: http://www.zoology.ubc.ca/biomath/.

Examination copies are available via the instructions here.

Princeton University Press author Sally Otto awarded a 2011 MacArthur Fellowship

Congratulations to Sally Otto who was awarded a 2011 MacArthur Fellowship. She is the co-author of PUP’s 2007 book A Biologist’s Guide to Mathematical Modeling in Ecology and Evolution. You can read more about her award here, or watch this video interview the MacArthur Foundation taped:

The award notice singles out the textbook and Sally’s dedication to educating others:

In addition to participating actively in laboratory and field experiments to test and refine her models, Otto is a dedicated educator, having recently co-authored an acclaimed textbook on mathematical modeling that introduces other biologists to the power and rigor of quantitative analysis. Otto’s extensive track record of bringing fresh perspectives to thorny conceptual problems suggests that her fundamental contributions to ecology and evolution will continue unabated.

Feedback Systems wins the 2011 Harold Chestnut Control Engineering Textbook Prize

The International Federation of Automatic Control just announced that the winner of the 2011 Harold Chestnut Control Engineering Textbook Prize is Feedback Systems: An Introduction for Scientists and Engineers by Karl Johan Åström & Richard M. Murray.

According to the IFAC’s call for nominations, this prize “is awarded to the author(s) of a control engineering textbook that has most contributed to the education of control engineers.”

Books are evaluated on their originality and innovation, their presentation of clear application to real problems, and in how well they meet education objectives. The prize (consisting of a monetary award, a certificate, and bragging rights for the next year) was formally presented at the Closing Ceremony of the 18th IFAC World Congress that was held in Milano, Italy earlier this month.

Feedback Systems is intended as a complete, one-volume textbook for undergraduate and graduate courses. A solutions manual is available upon request by professors who are assigning the book for courses and the authors are maintaining a wiki of additional content. A complimentary chapter is available for preview here. Instructors who wish to sample the complete book for their courses should follow the instructions here.

In addition to this prestigious award, Feedback Systems has received outstanding reviews in mathematics and engineering journals:

“This book provides an introduction to the mathematics needed to model, analyze, and design feedback systems. . . . Feedback Systems develops transfer functions through the exponential response of a system, and is accessible across a range of disciplines that use feedback in physical, biological, information, and economic systems. . . . Exercises are provided at the end of every chapter, and an accompanying electronic solutions manual is available.”–Mechanical Engineering

“[T]his is a refreshing text which is delightful to read, and which even experts in the area may find a valuable resource for its diverse applications, and exercises, and its clear focus on fundamental concepts that does not get side-tracked by technical details.”–Matthias Kawski, Mathematical Reviews

“This book provides an interesting and original introduction to the design and analysis of feedback systems. It is addressed to engineers and scientists who are interested in feedback systems in physical, biological, information and social systems.”–Tadeusz Kaczorek, Zentralblatt MATH

We hope you will join us in extending congratulations to the authors!

It’s official, e-Galleys are here, celebrate with a free book

Publishers Weekly ran a news item today confirming that Princeton University Press has signed up with Ebook Services to offer e-Review copies and e-Galleys of our books for review and possible course adoption. We are as excited about technology as anyone else and are doubly pleased that to celebrate our new partnership Ebook Services is offering complimentary digital comps of International Theories of Politics and Zombies by Dan Drezner to reviewers. This was one of our best-selling titles of the last year, so grab your freebie while you can (visit the Ebook Services site for details and limitations).

Ebook Services announcement: http://blog.ebookservices.com/2011/08/02/ebook-services-signs-princeton-university-press/
Publishers Weekly: http://www.publishersweekly.com/pw/by-topic/digital/content-and-e-books/article/48431-ebook-services-enters-e-galley-arena-with-digital-comps.html

PGS Dialogue: David Weintraub, author of How Old Is the Universe?

David Weintraub’s most recent book, How Old Is the Universe?, is a readable investigation of the title question that explains how we have arrived at an approximate age of 13.7 billion years for the universe. Weintraub works his way from biblical chronology of the origins of the universe to the high-tech astronomy research taking place today in this accessible and entertaining history. We recently posed some questions to Prof. Weintraub by email and are pleased to present this dialogue.

PUP: I am not an astronomer, so I was relieved to discover I could actually read How Old Is the Universe? You clearly went to great efforts to make the text accessible. How difficult was it to break down these big scientific ideas, terms, and facts for general readers?

Professor David Weintraub: Making sure I was speaking to a non-professional audience in English rather than in the jargon-filled language of astronomy was a constant challenge.  A major goal with this book is to help general-audience readers understand the complicated and unfamiliar concepts described between the covers.  Consequently I focused on this issue quite literally with every word I wrote.  At the risk of being struck down by the gods for hubris, I do think I have done better at this than most astronomers who are trying to communicate with a non-professional audience.  Nevertheless, more than a few of my descriptions passed through my ‘language of the lay reader’ filter unnoticed by me.  Fortunately, Princeton University Press assigned my manuscript to an editor who asked me lots of excellent questions for clarification, and quite often her questions arose when she bumped into a piece of text in which the meaning was unclear to her because of my too-technical word choices.  I do think, in the end, the presentation of difficult concepts in this book is accessible to the general reader because we paid such close attention to language and because I continually reminded myself of whom the readership of the book is intended to be.  My editor was a humanist who knew no astronomy before beginning to edit the book.  So she was my test reader; if she didn’t understand my words, I flunked the test.  When we were done, she felt that she understood every word and had learned and now understood everything in the book.  Fortunately for her and all readers, unless you are in one of my classes, there is no test at the end of each chapter.

(blog editor note: But if you do like homework, you can find lots of extra course materials on Prof. Weintraub’s site: http://sitemason.vanderbilt.edu/site/gVfcE8/How_Old_is_the_Universe)

PUP: One of the scientific principles you describe so clearly in the book is the rate at which the universe is expanding. While we are often told to assume nothing, you note that in this case, there is a lot of evidence to support the assumption that the expansion rate of the universe has been nearly constant. Can you talk a little bit about the expansion of the universe and the role dark energy plays in the process?

DW: The rate at which the universe has been expanding has changed over the lifetime of the universe, but since it first reached its current expansion speed it has not changed by much.  For the first few moments after the universe’s birth, the expansion rate was dramatically higher.  But that period of time (known as the inflationary epoch) lasted only a tiny fraction of a second.  Thereafter, the rate of expansion was nearly constant for ten billion years.  For the last few billion years, the expansion rate has been increasing, but the rate of increase is so slow, it is only barely measureable. Thus, it is safe to say that the expansion rate of the universe has been nearly constant for almost the entire history of the universe.

The actual measurements indicate we now live in an accelerating universe; these words simply mean that the rate of expansion is increasing. Dark energy is the name we give to the physical process that is causing the expansion rate of the universe to increase.  The evidence for the accelerating universe comes from measuring the distances and velocities (away from us, in all cases) of extremely distant galaxies; those measurements come from observing exploding stars called supernovae.  Supernovae are among the brightest objects in the universe; because they are so bright, we can detect them even when they are very far away, and because supernovae are stars that live and die inside galaxies, the supernovae are like flags that wave and tell us the distance to the galaxies in which they reside.

PUP: From exploding to imploding stars…In chapter four, you describe what happens to stars once they have drained the number of protons in their core. The resulting bodies are called “red giants.” What would happen to the Earth if (or when) the Sun turns into a red giant? How would it affect the other planets and bodies in our solar system?

DW: In about three billion years, the Sun will begin to evolve into a red giant.  So the question is not “if” but “when” this will happen.   During this red-giant phase of a star’s lifetime, the star puffs up.  The outer layers of the Sun likely will expand outward and fill up most of the volume of the solar system inside of the Earth’s orbit.  While the surface of the Sun will actually be cooler than it is now, it will still have a temperature of several thousand degrees, and that surface layer will be so close to the Earth that the heat received by the Earth will increase so much that the atmosphere and oceans will literally boil off into space.  The Earth itself might eventually turn molten before it is swallowed by the Sun.  Before the Earth is destroyed by the dying Sun, the planets Mercury and Venus, both of which are closer to the Sun than is the Earth, will suffer similar fates.  Mars is probably far enough away from the Sun that it might survive, but it also will lose what little atmosphere it has.  The outer solar system would be affected in less extreme ways.

PUP: This is the stuff of childhood nightmares, which leads naturally to childhood daydreams, or in this case, star-gazing. A good part of what we know about the universe, we know because of the close observation of identifiable stars. But there are hundreds of millions of stars; how is it possible to keep track of individual stars?

DW: Imagine sitting in the stands at a football game; you look across at the 40,000 fans sitting on the opposite side of the stadium from you.  You make a map of where each fan is sitting and identify each one with a section, row, and seat number.  With your binoculars, you carefully make some observations of each and every fan (long brown hair or bald headed? baseball cap or no hat?  glasses? beard? windbreaker or raincoat?) so that each person can be distinguished from his or her closest neighbors.  Now the rules for movement: no fan is allowed to move more than 1 inch per century.  Yes, per hundred year interval of time.  Now you go home, sleep like Rip van Winkle, eat, work, come back in fifty years and look again across the stadium.  You will be able to identify each and every football fan, both from their locations and their particular characteristics (provided they’ve been fed well and don’t age much). The stars are so far away from us that the rates at which they change positions are almost immeasurable, even in a human lifetime.  The stars do move, and astronomers can measure their motions, but their motions are so small that year after year we are able to easily find and re-find the same stars.

PUP: Another fascinating fact I took away from the book is that stars come in different colors. You note that some stars are blue and others are red, but that they may not be visible to the naked human eye. Why is this?

DW: Most stars emit light in fairly similar amounts across the spectrum of visible light (what our eyes can perceive) and thus appear white.  But red stars and blue stars do exist and are visible to the naked human eye.  Red stars emit light in all colors, but they emit more red light than any other color; similarly, blue stars emit all colors of light, but they emit more blue light than green or yellow or orange or red.

My eyes are not very sensitive to colors — I fail all the color-blindness tests — so to my eyes, red giants are barely different in color from most other stars (that appear white) and blue supergiants are only a bit bluer, as seen by me, than most other stars.  But to observers whose eyes are sensitive to subtle color differences, red stars and blue stars are easily distinguished from the rest of the stars.  Clear skies help; patience in observing the night sky helps; and knowing where to look helps.  But once a red star or blue giant star is pointed out to you, you would immediately recognize their colors.   Most of us simply have not spent enough time looking up at night, under dark skies, to notice the color differences among stars.

PUP: It seems as if any discussion of astronomy almost always leads to the great Hollywood question – “Is there extra-terrestrial life in the universe?” What are your thoughts on the possibility of finding ET?

DW: I find the universe to be incomprehensibly immense and lonely.  I almost long for the olden days — before Copernicus — when humans “knew” that the entire universe was small and we were of central importance to the workings of the universe.

I am not convinced, as most of my professional colleagues and Hollywood producers seem to be, that the universe must be populated with life, including intelligent life.  I also am not convinced that we are alone.  I think we lack the knowledge to make any claims, one way or another, about life beyond the Earth.  But I do find Enrico Fermi’s question “Where are they?” the one I think about the most.  Wherever they are, if someone else is out there, we haven’t found them yet and they haven’t found us yet and we see no evidence of their presence in or impact on at least our part of the Milky Way galaxy.  Our solar system is fairly hostile to life.  Venus and Mars, the most Earthlike planets in our solar system, did not survive for long enough as habitable planets for advanced life forms like us to make them home.  The Earth, in fact, may be a very special place.

This suggests to me that life may be rare, if not unique, at least in our part of the universe and that humanity must take more responsibility for ensuring the survival of life.  Not the survival of life on Earth but life itself.  We may have an incredibly important responsibility.  If we’re it, then if we screw this planet up, the grand experiment of life in the universe might end with us.

I do think that all the wild speculation about life beyond the Earth permits us to be very casual about life on Earth; it permits is to be cavalier about our stewardship of our planet.  Many religious beliefs also lead us to acting selfishly about life and our planet.  I don’t think such carefree attitudes about the health of our planet are good ones.

PUP: That segues somewhat neatly into my next questions. It may surprise some readers that you start your “popular science book” with an account of biblical chronology.  In fact, chapter two (titled “4004 BCE”) opens with a famous quote from The Annals of the World in which James Ussher determines that the universe was born on “the twenty third day of October in the year of the Julian calendar, 710.” Why did you select this quote? And why engage with a religious account that is at odds with scientific research?

DW: The quote, or at least a poor paraphrase thereof, from Bishop Ussher is well-known.  I placed an important quote at the beginning of each chapter that is related to the most important concept presented in that chapter, and I thought it would be of some value to place in front of readers an accurate quote from Bishop Ussher.

As for why the chapter is in the book, I wanted to recognize the role that biblical chronology and scholarship played in understanding the age of the universe.  For a short period of time, in the seventeenth century, scholars like James Ussher and John Lightfoot were at the intellectual forefront in this field.  Most scholars, let alone non-professionals, don’t recognize that fact and often laugh at this quote.  My goals with this chapter were to recognize the important role that biblical chronology played, as an attempt to answer questions about the age of the Earth and the universe, in the seventeenth century, but also to point out that this method of scholarship was quickly found to be flawed and thus was left behind.  As a scholarly discipline, biblical chronology never achieved scholarly success, and in addition, it was supplanted by modern science.  The history of science reveals progress in our knowledge; this chapter presents a good illustration of such progress.  Biblical chronology was cutting edge in 1650, but it was quickly supplanted by better ideas, scientific ideas.

My hope, with this chapter, is that I would engage some readers who might put some credence in biblical chronology as an accurate chronometer for the universe, but to do so without insulting them.

PUP: How did your fascination with astronomy begin?

DW: I have no idea.  I flunked out of boy scouts at age 11 because I failed, multiple times, the test for identifying constellations and simply gave up thinking I could ever achieve the rank of first-class scout.  I know that I was never interested in looking up and memorizing the patterns of stars.  But I was a youth and a young teenager when some big discoveries were made in astronomy — the cosmic background radiation in 1965, pulsars in 1968 — and I grew up in the early days of space exploration — the race to the moon, the Pioneer and Voyager missions to the outer solar system, the Viking mission to Mars.  So my youth was peppered with news and excitement about space.  I also read science fiction, and I know that certain stories (Arthur C. Clarke’s The Nine Billion Names of God, The Star, and 2001, foremost among them) had a big impact on me.  I found myself interested in what stars and galaxies are and how they worked, but mostly I found myself interested in planets and the question “how do planets form?”

PUP: What advice would you offer a person who is thinking about pursuing astronomy?

DW: Astronomy is fun and being an astronomer is a wonderful way to be spending my life.  I would encourage a young person interested in astronomy to follow that interest as far as it takes him or her.  Astronomy is a tremendously successful vehicle for getting our youth interested in science.  With apologies to my friends who are chemists, I think it is obvious that learning about black holes and dark matter is more exciting to most young people, at first blush, than studying about covalent and ionic bonds.  Once youth become excited about astronomy, however, they discover that they need to know some fundamental science and math in order to further pursue their love of astronomy.  As a result, it’s a short hop, skip, and jump into the serious study of mathematics, physics, chemistry, biology, and computer science, all of which are important as foundational disciplines for astronomy.  Most of those who start out interested in astronomy will discover that these or other related disciplines (geology, engineering) or more distant ones that they discover (neuroscience, anthropology) through their studies of these fundamental disciplines become their passion.

PUP: What are your hopes for those who read this book?

DW: I would like readers to share in the wonder of knowing that humans have discovered the age of the universe.  This is a phenomenal, almost outlandish achievement.  Rather than sitting back and saying “Astronomers claim this is so and I guess I should believe them,” I want readers to say “Astronomers have explained to me, step by step, how they obtained the age of the universe, and I understand what they have done and I agree with their answer.”

PUP: Do you have any future projects in mind? Perhaps another book?

DW: Yes, I am working on current projects and have future projects in mind.  My research, primarily on why very young stars produce X-rays and whether those X-rays can tell us anything about the formation of planetary systems around those stars continues.  But like all baseball players, I believe in jinxes.  So I would prefer not to jinx myself by telling you much more than that or by answering the second question.

Watch Prof. Weintraub’s recent lecture series hosted at Vanderbilt (Part 1, 2, 3, 4, 5, 6)

Nature’s Geometry

The geometry we learned in high school is ideal for describing “man-made” forms such as buildings, roads, fences, etc. But lines, circles, and triangles don’t seem to do justice to trees, clouds, or mountains. What about the forms of nature? Is there a geometry for them? The late mathematician Benoit Mandelbrot (1924-2010) pioneered just such a geometry; he called fractal geometry after the Latin word fractus, which means broken or irregular.A fractal is a shape composed of smaller copies of itself (think “fractured”). For example, a cauliflower is composed of florets—little flowers—which look just like little cauliflowers. We can use this idea to draw many natural forms using precise, step-by-step methods called algorithms. In the figure below we start with a simple, three-stick tree in (a) and then repeatedly turn each branch tip into a smaller, three-stick tree. The last step (f) is a computer rendering of the fractal the shapes are converging to.

Step-by-step drawing of a fractal tree.

The close-up below illustrates one of the reasons Annalisa Crannell and I chose the striking photograph Winter Road along the Trees by Wil Van Dorp for the cover of Viewpoints: Mathematical Perspective and Fractal Geometry in Art. The fractal beauty of the trees was impossible to resist!

Detail of the cover of Viewpoints.

Nowadays computers use fractal algorithms to generate photographically real landscapes in many feature films that require special effects. However, mathematicians and computer scientists may not have been the first to follow this road. As Benoit Mandelbrot pointed out, Asian artists have employed fractal-like portrayals of natural forms for centuries. As you can see below, Japanese woodblock artists of the nineteenth century used abbreviations for natural forms that are surprisingly similar to fractals investigated by mathematicians and scientists more than a century later!

Top: A “quadric Koch island” fractal as described by Mandelbrot.
Bottom: Boats in a Tempest in the Trough of the Waves off the Coast of Choshi (detail), from the series A Thousand
Pictures of the Sea
, by Katsushika Hokusai (1760-1849).
Top: Fractal generated by an iterated function system.Bottom: Shono: Driving Rain (detail), from the seriesThe Fifty-Three Stations of the Tokaido, by Ando Hiroshige (1797-1858).
> Top: Fractal model of two-fluid displacement in a porous medium.Bottom: Short History of Great Japan (detail), by
Ikkasai Yoshitoshi (1839-1892).


Marc Frantz holds a BFA in painting from the Herron School of Art and an MS in mathematics from Purdue University. He teaches mathematics at Indiana University, Bloomington where he is a research associate.

Annalisa Crannell is professor of mathematics at Franklin & Marshall College. She is the coauthor of Writing Projects for Mathematics Courses.




This is the final installment in a series of blog postings from the authors of Viewpoints: Mathematical Perspective and Fractal Geometry in Art.


Perspective by the Numbers as Art Appreciation

One of the best types of art appreciation course is a straight-up studio course in painting, drawing, or sculpture. Even a few lessons can provide a better grasp of the talent and discipline that go into the artwork we see in galleries and museums. But what about the contemporary art we see in movie theaters? More and more of what we see on the movie screen is computer-generated imagery (CGI), including entire films by the big animation studios such as Pixar, DreamWorks, and Industrial Light & Magic. Are there art appreciation lessons for this type of art?

It so happens there are. While a complete understanding requires a fairly advanced knowledge of mathematics and computer graphics, a good grasp of the basics requires only elementary mathematics and access to computer spreadsheet software. Annalisa Crannell and I devote a chapter to this in Viewpoints: Mathematical Perspective and Fractal Geometry in Art. One of my students, Tia, chose this medium for her final project. Although Tia was a biology major, you can see from the samples below that she was able to design a nice mathematical model of a lamp and lampshade, and use a spreadsheet to visualize them from any angle.

Part of a student’s final project

Tia’s project included multiple, hand-colored drawings made from scatter plots she generated in Microsoft Excel. This hybrid approach gives a good feel for the power of computers in 3-D imagery, without losing the connection between the relevant mathematics and the final artwork. The quality of her images underscores an important advantage of doing perspective by the numbers like this. Namely, it acts as a kind of safety net for people who lack talent or confidence in drawing, allowing them to make images that art majors would be equally proud of.


Marc Frantz holds a BFA in painting from the Herron School of Art and an MS in mathematics from Purdue University. He teaches mathematics at Indiana University, Bloomington where he is a research associate.



This is the fourth in a series of blog postings from the authors of Viewpoints: Mathematical Perspective and Fractal Geometry in Art.

Taking Heart, Making 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 friends confessing their fears of my subject. I understand that many people think math is hard and scary. That’s why I was so eager to do something easy and approachable, like drawing, in my math classes. I figured Viewpoints: Mathematical Perspective and Fractal Geometry in Art would allow my students to learn geometry by doing art.But to my great surprise, I found that it is the art, not the math, that makes people really nervous. As my coauthor 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. Sid’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.

Sid’s first drawing
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 flourished, and I was 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. Sid’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 he’s not going to be afraid of anything.

Sid’s final drawing

Annalisa Crannell is professor of mathematics at Franklin & Marshall College. She is the coauthor of Writing Projects for Mathematics Courses.


This is the third in a series of blog postings from the authors of Viewpoints: Mathematical Perspective and Fractal Geometry in Art.

Trying a New Viewpoint on Math

Perspective artists at Franklin & Marshall College


In the photograph above, Annalisa Crannell’s students learn perspective by tracing what one student sees through the window from a fixed viewpoint. The window on the far left contains a color painting of the same subject&#8212proof that the process really works. In grappling with perspective problems, our students have taught us a lot about creativity in math and art. On more than one occasion an art major (or an art professor in our workshops) has leapt to the blackboard to sketch a solution to a tricky problem in perspective, before many math majors made a mark on their papers. The artist might need help in proving the correctness of the solution, but it’s impressive that they guessed it so quickly. What’s their secret?A key issue is the difference in the way people view, and even define, the concept of a mistake. In art it’s commonplace to begin a project with many rough sketches, most of which are drawn very quickly. Although most of these sketches differ markedly in design and quality from the end result, they are viewed, not as mistakes, but as a natural part of the process. That’s how the artists approach math problems in perspective: by making quick sketches and approximations, until the beauty and symmetry of a solution suggests its correctness.

All too often, math students are afraid to make a mark on their papers unless they’re sure it’s correct. This is ironic, because professional mathematicians work more like the artists do: by making quick, rough guesses about what ought to be true, then proving or improving or discarding the result. They call the guesses “conjectures” rather than “sketches,” but it’s the same idea. Wading in and making mistakes is part of the creative process in any field. That’s why Annalisa and I took the approach we did in Viewpoints: Mathematical Perspective and Fractal Geometry in Art&#8212so that readers can learn not only what mathematicians do, but how they do it, and where the fun comes from.

Marc Frantz holds a BFA in painting from the Herron School of Art and an MS in mathematics from Purdue University. He teaches mathematics at Indiana University, Bloomington where he is a research associate.

This is the second in a series of blog postings from the authors of Viewpoints: Mathematical Perspective and Fractal Geometry in Art.


Looking at the world through new Viewpoint(s)

What are these people looking at?

When I started working with Marc Frantz 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. Above 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, 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 a 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 fence post?) 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.

Annalisa Crannell is professor of mathematics at Franklin & Marshall College. She is the coauthor of Writing Projects for Mathematics Courses.

This is the first in a series of blog postings from the authors of Viewpoints: Mathematical Perspective and Fractal Geometry in Art.