# Perspective in Math and Art

Teaching students about drawing changes their ideas about what they see and about right answers, writes Annalisa Crannell.

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.

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).*

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.*

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.

## Bio

*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.*

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Teaching students about drawing changes their ideas about what they see and about right answers, writes Annalisa Crannell.

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 Artand 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

Viewpointshasn’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.