Fractals: a pattern of chaos.
Though fractals are characterized by repetition, they also contain variations and differences. Even the slightest outside influence--such as a stream of warm air, a drop of rain, or the touch of a human finger--changes their outcome. Did you ever wonder why snowflakes are all different, yet contain distinctive, crystallized six-fold symmetry? That's because their shape is determined by outside influences on their tremulous journey through the atmosphere. These random influences are called chaos.
Good or Bad?
Chaos is neither good nor bad, but its random influences transform objects and create uniqueness. Just think how boring our world would be if every tree, every mountain, and every cloud were identical. What would our world be like if every person looked and acted alike?
Art or Science?
Fractal geometry was first posited by Benoit Mandelbrot, a researcher born in Warsaw, Poland in 1924. The word fractal suggests fractured and fractional. Fractal geometry focuses on wrinkled, broken, and uneven shapes--variations caused by chaos. Mathematicians model fractals by solving a set of equations and reinserting the solution back into the same set of equations. This process is called iteration.
Systems and objects in nature that are radically changed by chaos are called nonlinear. Artists have recognized and interpreted non-linear patterns for centuries. The work of artist Nachume Miller (1949-1998) is often interpreted as fractals. He said that "Looking at my work, you could see a seascape, a microscopic cosmos ... it could be the Milky Way." Fractal patterns have also been discovered in sounds of nature and music.
Computers and Fractals
Fractals can be created using mathematical equations and computers. Today, scientists use computers to better understand chaos and its laws within dynamic systems such as meteorology. However, they cannot always predict or exert control over chaos. A unique set of geometric equations, the Mandelbrot Set, express fractal patterns found in a variety of natural and humanmade objects. The repeated designs are easily recognized within geologic features, plants, and animals.
In your classroom, using a PC, a digital projector, and a freeware fractal generation program, select a fractal pattern and experiment by zooming in on chosen sections. Discuss how fractals repeat themselves as you continue to magnify the image. Use the drop fractal generation program to change the color schemes and fractal equation parameters to create additional patterns.
Remember to ask students to make connections and identify repeating shapes found in nature. Select student volunteers to make changes and create their own fractal designs for the class to enjoy. Copy the freeware onto CDs, and use them in your school's computer lab. Independent study will be a welcome event. Fractal print imagery could serve as a substitute for interactive software.
Divide the class into groups, and assign a disposable or digital camera to each group. Have each student record random patterns found in nature and humanmade objects. Remind them to look closely. After the film is developed or images are saved to a computer (and possibly manipulated), ask students to select one image that best reflects repetitive fractal patterns. Have students mount their images and write accompanying statements that explain the image content. The written component could be a haiku or other poem that serves as an aesthetic response.
Fractals are visual records of dynamic change over time. We see them every day within repeated forms, shapes, and lines of natural and humanmade objects.
Briggs, John. Fractals: The Patterns of Chaos. Simon & Schuster, 1992.
Students identify connections between the visual arts and other disciplines in the curriculum.
* In what ways are artists and scientists alike and different?
* What repeating lines, shapes, and forms can you identify in your environment?
* How do repetitive patterns and fractals make our world more interesting?
* How can we practice seeing fractals every day?
* How can our ability to recognize fractals help us as artists?
Can you recognize these fractals? (Answers, below)
a: What is it?
b: What is it?
c: What is it?
d: What is it?
a. hardened, unfinished concrete, 2 x 2'; b. thin glaze of ice on pond, 4 x 3'; c. basalt rock with lichen, 10 x 8"; d. weathered sandstone, 3 x 3'
Elisa Wiedeman lives in Flagstaff, Arizona. ECWiedeman@aol.com