I have always been the type of person who needed an explanation for everything. Nature has always been interesting and unexplainable to me. Why do oak trees and evergreens have different shapes? What is the shape of a cloud? How is it that the weather people on TV have difficulty making an accurate forecast? In the last three years I have started to find some answers as to why nature does not have the same predictability as many man-made events.

During the summer of 2002 I attended the Crash Course in Discrete Mathematics at Rutgers University in New Jersey. This course was created for teachers of Discrete Math and mathematics teachers that could use Discrete Math topics in their "regular" math courses. One of the topics covered was Fractal Geometry. This led to changes in how I teach "regular" math courses, how I work with my co-workers, and how I look at the world in general. One of the side effects is the project discussed in this article. I developed this project with Circe Dunnell, one of the art teachers at Rutgers Prep.

Fractals can be described as shapes with similarity of scales. Imagine a picture of a boulder, a rock, and a grain of sand. You can only tell the difference between the pictures if there were a frame of reference. Beniot Mandelbrot first used the term fractal in "The Fractal Geometry of Nature" in 1977. Mandelbrot said, "Clouds are not spheres, bark is not smooth, mountains are not cones, coastlines are not circles, nor does lightning travel in straight lines." Mathematically, fractals are based on iteration, which is each output becoming the next input. Traditional mathematics finds a new output for each value entered into the equation. Fractals are also connected to the study of chaos.

One of the ideas that Mandelbrot used with the study of fractals was the fractal dimension (Hausdorff Besicovitch dimension), or complexity. The ancient Greeks gave us an understanding of the first, second, and third dimension. The edge of an ice crystal is more "complex" than that of a stone or the edge of a leaf. If data points are graphed on log-log paper then the slope of the line is the fractal dimension of the object. This process is called the box count method of calculating the fractal dimension. (It should be noted that some traditional geometric objects, circles for example, do not have a fractal dimension because the data points on the log-log paper do not form a straight line.) Fractal dimension of computer generated fractals can be calculated using the formula N=(P/p)D. An object of unit size P is made up of N identical similar size units of size p with a fractal dimension of D.

I have been teaching fractal geometry during my Discrete math class since the 2002-2003 school year. During the spring of 2004 one of my co-workers, Gary Maas, gave me an article from the local newspaper. This article turned out to be one of the most influential articles I have ever read. A physicist at the University of Oregon, Richard Taylor, studied the fractal dimension (complexity) of a variety of pictures and found that they had a relaxing effect on people. The research found that pictures of natural objects, computer generated fractals, or the artwork of Jackson Pollock had similar complexity. Taylor found that pictures with similar complexity had the same effect.

I found this fascinating. As a math teacher I had never been in the art building on our campus, let alone understood what went on inside those walls. I approached Circe Dunnell and asked for her opinion on the article. Circe was also fascinated with fractals. We spent some time speaking about how we could get the students to work on fractals in both math and art class.

Before we went on summer vacation we decided that we would spend time over the summer discussing the project and working fractals into our curriculum. Although things were not as organized as we hoped, possibly chaotic, we started the year with fractals. Circe started the year working with her Portfolio 2 class, our highest level art course, working on creating work that was inspired by fractals. I started the year by teaching fractals in Discrete Math.

My Discrete Math course has two types of students. There are a few students that are taking an extra math course with AP Calculus and then there are the rest of the students that have hated almost every math class they have ever taken, but they are being forced into one more semester to fulfill their graduation requirement. We had five students that were taking Discrete Math and Portfolio 2 at the same time. The students were getting a double dose of fractals if they had both classes. I set aside time during the first few weeks to meet with the art students that did not have my math class.

I started the class by showing students examples of fractals. Some of the examples were assignments previous art students created (almost anything that comes from nature is going to have a fractal that you can talk about). Most of the examples were digital pictures of nature (trees, cracks in the ground, clouds, and water droplets on a glass top table) outside of my house. The math students spent time working on the formulas and mathematical aspects of fractals and the art students spent time trying to create their own fractals. Circe started her class in the same manner as I had by looking at and discussing fractals. The students discussed the compositional construction and inherent balance all fractals appeared to contain. In order to make certain all of the students in her class had a basic understanding of the mathematical principles behind fractals Circe had me speak to her classes at the outset of her assignment. As the students developed their work over the course of a few weeks, time was taken to discuss and critique the general direction they were headed in and the work wrapped up with a final critique of the overall composition. None of the art students were graded on their ability to create a fractal. They were graded on composition and other cool art stuff that I do not understand. The math students took a regular math test on the formulas and equations.

Later in the year Circe gave me digital images of the work the art students produced in that first assignment and I put them into a computer program that analyzes the fractal dimension of bitmap files. The program is called Benoit 1.3. The results were amazing. The decimal dimensions were between 1.5 and 1.8. The lower the decimal dimension, the smoother the line. But, the awesome part was that the standard deviation was less than .01 in all cases and .001 in one case. The students were incredibly good at creating a consistent fractal dimension in their work. The artwork included in this article are pieces that the students produced during the project. Dripping and using gravity instead of touching a brush or pen to the canvas helped the students create the consistent fractal dimensions.

Any teacher (I have done fractals with kindergarteners) can add fractal patterns to their math or art lessons. Young students can easily follow the patterns to create fractal images on paper. Older students can use Algebra 2 formulas to calculate shapes with zero area and infinite perimeter. Alex Barocas, an English teacher of seniors in our school, has a class entitled "New York: Art of the City" and he teaches a segment on Jackson Pollock and includes critical discussions of Pollock's work.

Since I started working with math and art together I see everything in a different light. I look at nature and see how the organization is similar in many areas. Photographs of river systems from outer space, the veins that can be seem on your wrist, and the branches on a tree look all look similar.

During the last few years I have spent many hours pestering Circe and the other art teachers discussing art and math connections. I now teach a class in Fractals & Chaos that includes a field trip to the Museum of Modern Art in New York. I have discussions with my students that would not have been possible a few years ago. Students talk to me about their art projects and we discuss ideas and the paths they are taking with their work. I have written college recommendations for students applying to art schools explaining the connections the students have shown between the worlds of math and art. I feel that my teaching has improved because I am able to recognize additional talents in my students. Fractals have given me a new way to connect with my students.

Mandelbrot published his work in 1977. In the mid-80's very few universities offered courses in fractals. In 2005 there were over 8000 research papers written using fractals. Google has 3,010,000 web pages and 141,00 images in a search for "fractal". This new branch of mathematics is going to be very important to future generations. In almost all aspects of life current events influence the events to follow. Nature is an iterated system.

Math and art do have places where they can work together. If you would like more information about the project or using fractals in your classroom then please email merges@rutgersprep.org.

Circe Dunnell was a great source information and ideas while developing the project and writing this article.

Bibliography:
Mandelbrot, Benoit. The Fractal Geometry of Nature. San Francisco: W. H. Freeman and Company. 1983

Taylor, Richard. Order in Pollock's Chaos. Scientific American. 287, 84.

Taylor, Richard, R. Guzman, T.P. Martin, G.D.R. Hall, A.P. Micolich, and C.A. Marlow. Authenticating Pollock Painting Using Fractal Geometry. 2004.

Kevin Merges is in his sixth year of teaching mathematics (mostly Precalculus) at Rutgers Preparatory School in Somerset, NJ. He taught previously at The Albany Academy in Albany, NY.

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