It has been a pretty busy last few days. Over the weekend, Thom and I started tackling our future collaborator’s nursery! We are so excited about how it’s turning out, we go back in a couple of times each day to relish in it. Very fun. We’re super psyched about the colors we have chosen, and the imagery that will go on the wall. I’m currently working on stencils for those images, which takes me back to my many years of doing installation work which required lots and lots of cutting with an x-acto knife. I’m really enjoying it (minus the occasional “flesh wounds!”).
I’ve also been sorting through all the baby loot that Nina gave me last week! Yikes! It’s funny to see all the month ranges that exist in little kids’ clothes: 0 to 3, 2 to 4, 3 to 6, 6, 6 to 9, 9, 9 to 12 and so on and so forth. I love to categorize things, but even that gets a little confusing. But in the end, it’s getting done, and it makes me feel quite accomplished! Ha! Ha!
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.
This summer, Thom and I spent a long weekend in Lancaster, PA. On our way to lunch the first day, we saw the entrance to Franklin & Marshall College and it made us curious to know what type of school it is. Thanks to our trusty iPhone I did a Google search right away…what’s interesting is that on the College’s home page there was a slide show of stories, one of which, coincidentally, happened to be about Annalisa Crannell’s involvement in this research about combining math and art. As you can imagine, I was immediately hooked, and I just knew that once I’d get home, I’d contact her and Marc about doing a “why people draw” interview.
Lucky for me, both Annalisa and Marc were happy to participate, and I’m so glad they did. Even though they did this interview separately, they were able to establish a really nice conversation between the two of them through their answers. For those people who love math, as well as for those who might not care for it as much because of past experiences, you’ll find that their approach to math concepts and their use of drawing as a way to facilitate understanding is such a welcome idea. I also hope that there are teachers and parents out there reading this who will be inspired to go deeper into this reality that drawing can have a place in all classrooms, not just the art room!
ANNALISA CRANNELL & MARC FRANTZ (tah-dah!)
So, without much more from me, here they are (drum, drum, drum) Annalisa Crannell and Marc Frantz in their own words and images. Thank you Annalisa and Marc for taking time to share such interesting ideas as part of the “why people draw” series!
HW: What’s your earliest memory of drawing (of being able to draw)?
AC: Actually, art doesn’t figure very large in my early memories.
I’m sure I drew a lot as a young child, but my first real memory of drawing comes from a 7th grade art class. I was with one of my smart alecky friends, Mary. The teacher gave us a pre-test with the question, ”Can you name any Renaissance artists?” and Mary’s one word answer was “Yes”. (The teacher changed the wording of the question on the post-test). I remember the song “She’s a brick House” blaring on the radio in that class. I also remember drawing a geometric design once, but I don’t remember any of the other drawings we did.
I seemed to move into drawing in my professional life without any real awareness of my own drawing biography.
MF: During my first years of school, my dad was training to be a programmer on one of the early business computers. He would bring home reams of typing paper for my brother and myself to draw on, and he would let us use his flow chart template; it said “Remington Rand Univac.” It was ideal for drawing spaceships. I used the Display symbol for a space capsule; the start/end symbol for elongated fuel tanks; the On-Page Reference symbol for spherical fuel tanks, the Manual Operation Symbol for rocket nozzles, and the Data symbol for fins. With these, even a little kid could make great-looking drawings. That’s when I first realized the power of algorithms in drawing. (An algorithm is a process that is done the same way each time, like long division in math, or dry-on-wet in watercolor.)
HW: What sparked the idea to create a course and workshops combining art, drawing and mathematics?
AC: Indirectly, the reason is that I was trying to find ways to engage my students with problems that were more immediate and tangible than the usual “When do truck A and truck B meet?” questions. Lancaster, Pennsylvania is rich with quilts that have wonderful mathematical structures sewn into them, so quilts make a great subject for a math class. Constructing these quilts requires a deep, if intuitive, sense of geometry, algorithm, and symmetry; each of these aspects has profound mathematical meaning. (The quilts are pretty to look at, too! I wonder if the people who made them thought about that? Hmm…)
I wound up in Indiana on one of my sabbaticals at the same time that Indiana University was applying for an NSF grant. Their big idea was to have mathematicians partner with people from other disciplines to do mathematics in context. I was already hooked on the idea, and I latched onto a brilliant guy named Marc Frantz. He’d already gotten an MFA from the Herron School of Fine Art and a Masters in Mathematics. Together, we started working on a course.
Pretty quickly, I switched over from quilts to perspective. I was following Marc’s lead: he realized that there was REALLY neat mathematics there that few mathematicians knew about, and I was intrigued.
MF: When Annalisa was visiting my school (IUPUI) on sabbatical, I got curious and began poking around the Franklin & Marshall College website. I saw that she regularly taught mathematics courses that required students to write, and write well, about what they did. (If you can’t explain a math problem in words, then you probably don’t understand it.) Her sample problems and advice about writing were spot-on.
I was impressed that someone could design a course so creatively, and envious that they would even be allowed to. Then from out of the blue, Annalisa and I were asked to create a math and art course as part of a big, National Science Foundation-sponsored project.
Filled with adrenaline, I began by revisiting some ideas that had worried me in art school. We learned perspective drawing as a bag of tricks, but with very little understanding (at least on my part). When I began reviewing these tricks, I realized that there was a lot of fascinating math behind them. We began formulating drawing problems that could be used as carrots to entice students to learn mathematics.
HW: What are some of the things you have learned about the human process of drawing and seeing throughout your research and teaching?
AC: One of the big things I’ve learned is that drawing takes time. And that’s good. It slows you down and can be meditative. Not many people think that solving algebra problem after algebra problem is much fun, but drawing many, many windows in a building is a satisfying kind of repetition.
Drawing helps my students learn to do what I do with my own mathematical research. When I’m stuck on a problem, I keep drawing picture after picture, hoping that one of them will be the construction that explains the underlying situation in a way that I want. It’s only after I’ve drawn a gazillion pictures (is “gazillion” a highly technical mathematical term?) that I finally “see” the answer. Then I can try to explain it to others. In the same way, my students learn to spend a lot of time drawing and drawing cool perspective pictures, and by doing this they internalized the underlying geometry.
MF: The insights that have most influenced the way I see the world have come from fractal geometry. Michael Barnsley began his textbook Fractals Everywhere by saying, “Fractal Geometry will make you see everything differently. There is danger in reading further.” He wasn’t exaggerating. Once you’ve learned about fractals—shapes that are made of smaller copies of themselves, like trees are made of branches—there’s no going back. After that, you see fractals everywhere: in trees, clouds, mountains, lightning, even in the kitchen (cauliflower and broccoli). Interestingly, the artists beat the mathematicians to this discovery. Asian artists, particularly Japanese woodblock artists in the nineteenth century, often formalized natural shapes as symbols virtually identical to fractals studied today.
HW: What is your definition of drawing? Has your definition changed after your research? If so, in what way(s)?
AC: I don’t think I want to define drawing, but I can describe what drawing means to me. For me, the act of drawing is a way of exploring ideas, and a finished drawing is a way of conveying ideas. These ideas could be a sense of space (as in many of my students’ perspective drawings) or a mathematical theorem (think about drawing the Pythagorean Theorem) or a sense of structure (as in drawing trees or mountains using fractal algorithms).
I don’t know that my research has changed the meaning that drawing has for me, but working with Marc has certainly enhanced my appreciation of drawing as a powerful intellectual tool.
MF: Sometimes a drawing is referred to as a “study.” I feel now more than ever that drawing and studying are inextricably linked. When I want to understand something better, whether it’s something in the real world or an idea in mathematics, I almost always take pencil and paper and make a drawing of some kind. That’s just my way of relating to things and ideas.
HW: In what ways has your own appreciation for art been enhanced by teaching a mathematical approach to looking at it?
AC: I think I described above the appreciation I have for the act of drawing. Teaching perspective from a mathematical point of view has also taught me how much more I can get from a painting by looking at it from many distances: getting up close to look at the brush strokes, getting way far back to see the whole composition, and also finding the perfect middle ground where the painting “pops” into perfect perspective, where I feel like I’m actually standing in that cathedral pictured on the canvas.
teachers in a gallery mentally tracing appropriate lines by holding up shish kebab skewers during a "mathematics & art workshop" (photo courtesy marc frantz)
MF: I agree with Annalisa that it’s a revelation to see a good work in perspective, viewing with just one eye from the viewpoint—the place from which the painting has an incredible sense of depth. One of my former art professors said she was in a gallery in Italy that had tiles on the floor to show you where to stand in order to do this. For the most part, however, the experience is a well-kept secret, as are the geometric techniques used to find the viewpoints. I never get tired of treating friends to a viewpoints tour of my local art museum, and hearing their gasps of astonishment. That’s where the title of our book came from.
once the viewpoint is determined, students can enjoy the painting from that spot, as one of marc's former classes did at the indianapolis museum of art (photo courtesy marc frantz)
HW: Could you explain, in layman’s terms, how projective and fractal geometry relate to drawing?
AC: “Projective geometry” describes how we project the three-dimensional world around us onto a two-dimensional canvas. Cameras do this all the time, and they can do it with any shape, including a human face. Projective geometry handles mostly the part of the world that is points and lines. Because of that, my students draw a lot of man-made objects: buildings, letters, furniture, etc.
“Fractal geometry” is better at dealing with natural objects, particularly objects with neat textures. The word “fractal” comes from the same root word as “fracture”; it describes things like broccoli or clouds, where if you break a little piece off, that piece looks like the whole thing. Fractal drawings contain a lot of repetition at smaller and smaller scales, like drawing a big tree that has branches, each of which has tinier branches.
MF: I couldn’t do better than Annalisa’s explanation.
HW: Could you describe the objective and the process of doing tape drawings on windows that you ask your students to do?
AC: I could stand up in front of my students and say, “When you draw a picture of lines going away from you, you have to use a vanishing point.” But that’s not the best way to teach: it’s boring, and the students think I’m telling them some weird, arbitrary rule.
Instead, Marc and I have our students work in groups to create a drawing of what they see in the world. One student (the “art director”) stands about 4 to 6 feet from a large window. The other students put drafting tape on the window where the art director sees edges of buildings and other objects outside. From the art director’s position, the tape lines up exactly with lines in the outside world. When the students are finished with their picture, I can show them that THEY created this thing we’ll eventually call the vanishing point. So they are telling me how to draw the world; I’m not telling them.
student as "art director" standing in front of windows, rooted to the spot, with one eye closed (photo courtesy marc frantz)
student "art director" directs their team of "artists" to outline on the window the view of the world they see with their one, immobilized eye (photo courtesy marc frantz)
We come back to the window-taping day over and over in our class. Almost everything else we do builds on their own experience that day, not on some textbook definitions and rules of perspective.
"drawing" with masking tape to help make straight lines and minimize the mess (photo courtesy marc frantz)
a "sketch" marc made with a former student (photo courtesy marc frantz)
HW: What gets your Hamster Wheel running? (What gets you itching to draw or create?)
AC: I start drawing when I want to solve a problem. For example, right now I’m trying to understand how a thing called a “projective colineation” can come from a series of things called “perspective colineations”. I’m really, really stuck—and that’s actually sort of fun. I’m drawing lots of pictures to try to get unstuck, and by drawing these pictures I’m learning more and more about the problem. The pictures aren’t beautiful in the usual sense, but for me they contain some deep mathematical meaning, and that meaning is important to me.
projective colineation (courtesy annalisa crannell)
projective colienation (courtesy annalisa crannell)
MF: At this point in my life drawing is more fun if it involves mathematics in some way. I often get ideas for my research by drawing diagrams of mathematical ideas. Sometimes a line of investigation doesn’t seem to involve drawing, but then a drawing gets me unstuck at some critical point. And whenever I’m writing for an audience of more than just narrow specialists, I try to summarize the main points using pictures of some kind. Not all mathematical ideas involve pictures, but I’m extremely partial to those that do.
Thanks again Annalisa and Marc! May drawing continue to be the tool that helps you get “unstuck”!
For more information on what Annalisa and Marc have researched about drawing and mathematics, please be sure to visit the resources below.
Annalisa Crannell at Franklin & Marshall College
Mathematics of Art course description
Writing in Mathematics course description
Bryn Mawr Now
Mathematical Association of America (MAA) Distinguished Lecture
Inside Higher Education: Perspective in Math and Art by Annalisa Crannell (July 18, 2011)
Society for Industrial & Applied Mathematics (SIAM): How to Look at Art by Marc Frantz (May 14, 1998)
Drawing with Awareness (April 2003)
What I Wish I Had Known in Art School: Foundation of a Course in Mathematics and Art
Princeton University Press Blog
Princeton University Press on Facebook
A Challenge from Annalisa Crannell
Viewpoints: Mathematical Perspective and Fractal Geometry in Art written by Marc Frantz and Annalisa Crannell
Viewpoints: Mathematical Perspective and Fractal Geometry in Art written by Marc Frantz and Annalisa Crannell (Google Books)
Writing Projects for Mathematical Courses written by Annalisa Crannell, Gavin LaRose & Thomas Ratcliff
Starting our Careers written and edited by Curtis D. Bennett, edited by Annalisa Crannell
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