A picture indeed may be worth a thousand words, but for students participating in one National Science Foundation (NSF) Research Experiences for Undergraduates (REU) program through the Texas A&M University Department of Mathematics, it's the sum of thousands of individual coordinates that add up to one unique work of art.

For nearly eight weeks this summer, visiting students from across the nation met daily on the Texas A&M campus and meticulously worked to hand-color a series of grids -- each consisting of a hundred pixels -- that together formed a giant image of recursive, non-intersecting shapes otherwise known in mathematics circles as a "fractal dragon" due to its finished dragon-like appearance. The project is one of several research-based activities featured within the program, which is designed to incorporate educational and team-building skills to benefit students as they progress in their college and professional careers.

"Apart from undergraduate research experience, we try to create positive social interaction," said Dr. May Boggess, senior lecturer of mathematics and program coordinator. "The fractal coloring is just one of the recreational things we do."

The fractals are actually smaller sections of a larger, more complex fractal. Not easily explained by classical geometry, a fractal is an irregular geometric shape that can be divided into sections, each of which is a reduced-sized version of the original image -- a property called self-similarity. Fractals have a variety of applications, from computer graphics to textile art. Boggess's students, who are part of a bigger REU program on Mathematical Modeling in Ecology and Physiology, worked with a general class of fractals known as the Mandelbrot Set, named after famed mathematician and discoverer of fractals, Benoit Mandelbrot.

The fractals are brought to life via a pain-staking process that begins with Boggess pinpointing a visually appealing section of the Mandelbrot Set. Then, a computer code program she designed calculates the image's convergence properties and its particular color at each point on a gridded plane. The entire grid is converted into numerous 10-by-10 sections of rows based on the first letter of the color that goes in that particular spot. It is essentially a coded map indicating specific colors and positioning required to form the image. The sections of grid then are colored and pasted on a foam board, ultimately forming the resulting fractal dragon.

After spending several weeks on the artwork, REU student Sara Krueger, a senior biology and mathematics major at Bethany Lutheran College in Mankato, Minn., left with a greater appreciation for and a better understanding of fractals, which Boggess notes often occur in nature. Snowflakes, clouds and even broccoli, for instance, are among the many examples of things known to naturally form the unique geometric shapes.

"I'm very excited to see what the final fractal will be, because it is very hard to picture what it will look like," Krueger said. "It's a very interesting topic of mathematics."

Before progressing to paper or pixels, Boggess first piques her students' interest in their subject by showing them a documentary called NOVA: Hunting the Hidden Dimension that explains the history of fractals, how they are generated and their numerous applications. After dedicating an entire evening during the second week of the REU program to coloring each section according to its prescribed coded-letter pattern, the students were able to complete roughly a third of the coloring. The remainder was completed during the students' free time in the subsequent weeks of the program, resulting in the finished product that, once framed, will be displayed in the department's conference room in the John R. Blocker Building alongside four other fractal dragons created by previous participants in the program.

Patrick Davis, who graduated this past spring from Eastern Michigan University, was a senior majoring in mathematics when he participated in Boggess's program in summer 2010. Davis utilized his acquired knowledge of fractals as well as the entire REU experience as a precursor to graduate school, where he is working toward a doctorate in applied mathematics.

"I was able to learn quite a bit about fractals from this project," Davis said. "Before this project, I only had a minimal knowledge of what fractals are -- not really much more than what they look like. The film and our discussions really helped me to better understand what fractals are and how they are used."

For the past 12 years, Texas A&M's Department of Mathematics has played host to a variety of REU programs, which are funded by NSF to enable small groups of undergraduates from out-of-state colleges and universities to work in research programs at a host institution in hopes of encouraging them to pursue graduate degrees. Aside from research projects, Boggess said her students also participate in field trips that allow them to experience Texas culture firsthand. Students are granted stipends for their involvement with REU, and in many cases, assistance with housing and travel during their stay.

"We get really excellent students," Boggess noted. "A lot of these kids are coming to Texas for the first time, so they are really excited to be here. We want to show them a good time and help them experience Texas."

To learn more about Boggess's REU program and fractal dragons, visit http://www.math.tamu.edu/~mboggess/FRACTALS/fractal.html.

For more information on REU programs in the Department of Mathematics, go to http://www.math.tamu.edu/undergraduate/research/REU/.

To learn more about undergraduate research activities in the College of Science, go to http://www.science.tamu.edu/research/REU/.


Contact: Chris Jarvis, (979) 845-7246 or cjarvis@science.tamu.edu or Dr. May Boggess, (979) 862-4190 or mboggess@math.tamu.edu

Jarvis Chris

  • Art of Approach

    Texas A&M's Dr. May Boggess (left) and her students, including Myrielle Allen-Prince (right) from Bennett College in Greensboro, North Carolina, enjoy coloring for an educational cause as part of a unique Research Experiences for Undergraduates (REU) program in the Department of Mathematics.

  • Byte by Byte

    The process begins with a computer-coded program designed by Boggess to calculate thousands of specific positions and colors for each pixel point on a massive gridded plane made up of hundreds of 10-by-10 sections.

  • Points of Interest

    Each pencil used to color the 10-by-10 squares is labeled to correspond with a coded map that, if executed correctly, will result in an overall image known as a "fractal dragon" because of its completed appearance.

  • Group Synergy

    The latest fractal dragon begins to take shape.

© Texas A&M University. To request use of any of our photographs for educational use or to view additional options from our archive, please contact the College of Science Communications Office.

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