The Frame-Work of Fractals

Michael Frame walked up to the podium, carrying his transparen­cies, mentally going through his presentation. When he looked down at the crowd, he was shocked to see one particular person in the audience: Benoit Mandelbrot. Frame was to give an introductory speech about Mandelbrot, with the legendary mathematician sitting before his very eyes.

This all started back when Frame was a professor working on differential topology at Union College. At the time, Frame had been collaborating with physicist David Peak teaching an introductory course on fractals. A few years later, the nearby State University of New York at Albany was presenting an honorary degree to Benoit Mandelbrot, father of fractal mathematics and Yale Sterling Professor Emeritus. Peak was asked to give an introduction speech at the ceremony, but he was out of town that weekend. Instead, the offer went to Frame.

Looking back, Frame says it was like “giving an intro to the ten commandments with Moses sitting in the front row.” After­wards, Mandelbrot complimented Frame’s talk, mentioning that the two should work together in the future – a gesture Frame dismissed as politeness on Mandelbrot’s part. To his surprise, two years later Frame received a call from Mandelbrot, asking him to come to Yale to teach a course on fractals.

Fractals belong to a class of geomet­ric structures that contain a pattern that repeats itself at smaller scales. When sub­parts resemble the whole, the structure is called “self-similar,” and self-similar fractals have an irregular and fine structure at all scales of measurement. Fractal self-similarity and irregularity are seen often in nature, such as in leaf veins, tree branches, tributaries and mountains.

Making Math Digestible: Guidelines for Excellent Teaching

After Frame received the invitation from Mandelbrot, he alternated semesters at Yale and Union College, teaching the introduc­tory fractal course at Yale. Since then, he has expanded to teaching seven math courses, including MATH 190 and MATH 290 (Introductory and Intermediate Frac­tals) and Math101 (Geometry of Nature).

Even at Yale, where “you have Field medalists teaching undergraduates,” it is sometimes unusual for a professor to focus so much on undergraduate teaching. When asked about it, Frame responded, “When I started teaching, I just got hooked.”

Frame now holds Yale’s Dylan Hixon ’88 Prize for teaching excellence in the natural sciences, awarded to him in 2009, and the McCredie Prize for the best use of technology in teaching at Yale College, awarded in 2003.

Frame’s first teaching experience was in graduate school. As a teaching assistant (TA), he led recitation for a multivariable calculus class and ended up taking over the course when the professor became ill. At that point, he had no idea what the students’ level was, but he remembers thinking, “All they are going to get is going to be from me, from the book, or from talking to each other.” As a result, he tried to make as the subject as easy as possible for the students by starting with the simplest concepts and building up and presenting straightforward explanations.

Frame advocates a simple guideline for teaching: teach in a way that aligns with your personality. He explained, “For example, if you’re a nice guy, don’t try to feign tough­ness.” He continued, “For me, because I’m not the brightest guy, this means really simple explanations. I understand what it’s like to not understand something.”

That might be a surprising statement to hear from a Yale math professor. When probed further about it, Frame just laughed, “I’m the stupidest guy in the department… the dimmest bulb in the pack here. My stuff is a lot simpler, so if I’m going to make a contribution to the department, it will not be by writing papers that astound the world. It’ll be by teaching.”

This sympathy for students drives Frame’s love for teaching. “Teaching means having endless office hours and writing syl­labi and being as patient as you possibly can. When the light goes on in their eyes, that’s the moment that makes it all worthwhile.”

Searching for ways to make the learning easier, Professor Frame puts an incredible amount of planning into the content of his classes. He likes to mix in practical applica­tions and historical development along with the theory in order to provide a context that makes the math more interesting.

For example, when introducing geom­etry, rather than bombarding students with mathematical definitions and triangle theorems, Frame presents the following problem: “You’re on the island of Samoas, and you want to build an aqueduct through the lake, starting from two towns. How do you make sure you meet in the middle?” This simple example shows a powerful application: the creation of a third observa­tion post against which to reference angles.

When discussing trigonometry, Frame explains to his students that the Greeks cal­culated the size of the earth, distance to the moon, and other astronomical figures using only trigonometry. “When I tell this to a class, they’re flabbergasted. ‘The Greeks knew that?!’” he said. Frame accounts for the success of this method by explaining that the students are focused on the stories and applications, with learning a great deal of math a side effect.

Frame also draws many connections to other “less frightening” subjects, in the hopes of better engaging his students. He discusses the geometry of snowflakes and honey combs. He frequently draws allu­sions to art, such as the mathematical logic behind Escher’s optical illusions, or the four dimensional hypercubes in Dali’s Crucifixus (Corpus Hypercubus).

Frame employs other tactics to make math more digestible. He makes extensive use of his class websites to complement lectures, which is particularly invaluable for fractals classes that are “so image heavy.” He even tells jokes in class. “They’re truly terrible jokes, and they’re effective because of it,” he said. “It makes the math seem not so bad afterward.”

Constructing Fractals Beyond the Classroom

Frame’s course offerings in fractals mirror his research interests. He has worked with iterated function systems (IFS), a method for generating fractals that starts with some image and a set of functions (transformations such as rotate, scale, and shift) and then applies all the transforma­tions to the image, over and over again (see Figure 1). The fractal is the final structure that satisfies the self-similarity property for this set of transformations. Apply the transformations on any other image, and something different is produced, yet apply the transformation on the fractal, and the fractal itself is the result.

Figure 1. Applying the same transformations iteratively to the starting image of a cat creates a fractal.

The same image is generated by applying the transformations one at a time in random order. An illustrative example is the four corners chaos game. Here’s how it works: start with a point at the center of the unit square, and choose any of the four corners. The new point is the halfway point between the old point and the corner. In this case, each transformation corresponds to a dif­ferent corner. If repeated, a filled-in square is created. Performing the same experiment with the corners of a triangle, however, does not produce a uniformly random filled-in square. Instead, it produces a fractal (see Figure 2).

Figure 2. The chaos game played with four corners creates a randomly filled in square (left). When played with three corners, it creates a fractal (right).

Iterated function systems are more than just pretty pictures; they have practical applications as well. For example, IFS can be used in statistical analysis, where instead of choosing the transformation randomly, you can base it off a stream of data, such as DNA sequences or cardiac rhythm data (see Figure 3). The resulting image is then representative of the data, and correlations can be readily visualized, giving IFS an advantage over standard statistics.

Figure 3. The four corners chaos game used to analyze data. Left: driven by DNA sequences (each corner corresponds to a DNA base). Right: driven by cardiac data.

Furthermore, IFS can be made much more complex. For example, a variety of functions can be used (such as a combina­tion of a shift and a rotate), each transfor­mation can have varying weights of being chosen, or some sequences of transforma­tions are not permitted.

The last complication, restricting the allowed transformations based on the currently selected transformation, drew Frame’s interest. To illustrate the prob­lem, consider the chaos game with a new restriction: corner 4 cannot be chosen immediately after corner 1. In this case, a new fractal is produced, which looks similar to the original except for the fact that the squares are more focused toward corner 1, shifted away from corner 4 (see Figure 4). Such a system is known as a memory restricted IFS because at each step, the transformations “remember” and are lim­ited by what happened on the previous one.

Figure 4. The chaos game with memory restrictions. The leftmost image has the restriction that the upper right corner (corner 4) cannot be chosen immediately after the lower left corner (corner 1). This causes the density of the points to shift toward the lower left.

Frame wondered whether the images produced by these memory restricted IFSs could be reproduced with a different set of functions, without any memory. Many years earlier, Jennifer Lanski, one of Frame’s students, had the same question. Working together, Lanski and Frame found simple conditions that determine when an image generated with one step of memory can be generated without memory. The left and middle images of Figure 4 can be generated without memory, while the right cannot. The natural follow up question was, is the same true regardless of the number of steps in memory?

As Frame began working to answer this question, he ran into difficulties and soon all but gave up on it. Then, nearly ten years later, the answer seemed to come in an instant. He described his bewildered reac­tion: “how come I didn’t see this 10 years ago?” Frame and Lanski had expected that the solution to the original problem could be extended and that there must be more restrictions, but it turns out that none exist.

The Pleasure of Problem-Solving

Frame, like many others mathemati­cians, speaks of the beauty of problem solving. “The pleasures of doing science are unequaled in the world. Somehow the creativity has to mesh with logic, so it’s restrained in a way, but in the end you get wonderful stuff.” He continues, “To some extent, it is the same as the pleasure of teaching – when the light goes on.” As a person who is quite familiar with difficul­ties in math, Frame understands the intense satisfaction of that “Eureka!” moment. Seeing it in his students is just as rewarding.

In chaos theory, there is a well-known idea called the Butterfly Effect. Named after the notion that the flapping of a but­terfly’s wings can unpredictably engender huge changes in future weather, the Butter­fly Effect is the principle that tiny variations in initial conditions can lead to big changes in a dynamical system over long periods. Frame’s story of speaking at Mandelbrot’s ceremony is a perfect example to illustrate that principle. As Frame puts it, “If I had never worked with Peak, if he weren’t out of town that day, if I never gave that talk, if I never met Mandelbrot – then I wouldn’t be here today.”

About the Author
SHERWIN YU is a sophomore in Morse College, majoring in Molecular Biophysics and Biochemistry and Computer Science and Math.

Further Reading
Mandelbrot, Benoit B. and Frame, Michael L. Fractals, Graphics, and Mathematics Educa­tion. Mathematical Associaton of America, 2002.