Mesmerizing Fractals
Summary
Introduction
Do you ever wonder what mathematicians study, and why? Most of what they do is complex and inaccessible to laymen, but fractal art might give us a glimpse. While mathematicians study fractals formally, they are used in many branches of science and technology, and we, as laymen, are struck by their beauty. Are you eager to create one? In this activity, you will get to take out some finger paint to make artwork and discover how common fractals are.
Background
Fractals are geometric figures, like triangles and circles. They are hard to define formally, but their features and beauty make them accessible and intriguing.
One feature is self-similarity, which describes how fractals have patterns that recur at different scales. In other words, when you zoom in, you will find a smaller version of a pattern you had seen initially. When you zoom in some more, you will find an even smaller version of that pattern, and so on. This seems to go on infinitely.
The way fractals scale is another feature that sets them apart from traditional geometric figures like lines, squares, and cubes. When you double the length of a line segment, the original line segment will fit twice (or 21) in it, making a line one-dimensional. If you double the length of the sides of a square, the original square will fit four (or 22) times in the new, bigger square, making a square two-dimensional. Do the same on a cube, and the original cube will fit eight (or 23) times into the new figure, making the cube three-dimensional. Apply this operation on a fractal, and the number of times the original fractal fits into the bigger one could be 3 or 5 or any other number that is not a whole power of 2. This is characteristic for fractals. They have a fractional dimension, like 8/5.
A last surprising fact: some fractals can show an infinite perimeter, while their area is finite. This is something you can explore in the artwork you are about to create.
Materials
- Old newspaper to protect your work area
- Old CD jewel case
- Finger paints
- Paper towels or cloth
- Water
- Paper to make a print; preferably finger paint paper
- Optional: magnifying glass
Preparation
- Protect your workspace
- Disassemble the CD jewel case so you have two loose covers.
Instructions
- Place one of the covers on your workspace with the flat part facing up, with the edges of the cover below it.
- Add a blob of paint about the size of a grape onto the center of the cover.
- Place the second cover on top of the first cover so the paint gets squeezed between the flat areas of the two covers and the edges of the covers face out. What do you notice? Does the paint spread? What type of geometric figure does it form?
- Look at the perimeter of this figure. Is it way longer or shorter than the perimeter of the CD cover?
- What happens when you lightly press the two covers together, and then release again? Does the geometric figure change? Do patterns like cliffs or other branching patterns form?
- What do you think will happen when you peel the top cover off? Will the figure change? Will it branch out even more? Will you get two figures, one on each cover? If so, will they be identical?
- Place the cases back down on your workspace and carefully lift the top cover off. Try not to slide the covers over one another, but lift it straight up.
- Look at the resulting figures. Were your predictions correct?
- Choose one figure and find a pattern. Then, see if you can find a bigger or smaller version of this pattern in your figure. Mathematicians call figures or patterns that are replicas of each other, but where one can be bigger or smaller than the other, “similar” figures or patterns.
- Place your cover, flat side down, on the paper to form a print. Do you think the printed figure will be an exact replica of the figure on the CD case? Why or why not?
- Peel the cover off. Again, avoid sliding the cover over the paper. Was your prediction correct? Is your printed version more detailed, or less?
- Fractals are geometric figures with patterns that repeat at several levels of magnification. Can you find smaller and bigger versions of a pattern in your figures? If you can, you just created fractals.
- If you have a magnifying glass, use it to zoom in and see if you can find the same patterns at an even smaller scale. Patterns that reappear at smaller and smaller scale is a typical feature of a fractal.
- Look at the perimeter of your fractal. Imagine placing fine twine all around your figure, matching the smallest details of its edges and cutting the twine so it can just cover the perimeter of your figure. Then, imagine stretching out this piece of twine. Would it be way longer or shorter than the perimeter of your CD cover? Now, imagine the fractal would branch off into smaller and smaller details. What would happen to the perimeter of the fractal? Would its area grow as fast? Would this fractal still fit on the CD cover, or would it need a bigger surface?
- Make and study some more prints. When you are done printing, clean the covers and repeat the process by adding a new blob of paint. How is your new figure like the first one, and how is it different? Try combining different colors and explore what yields the most beautiful fractals.
- What do these prints make you think off? Do they resemble something you know?
Extra: Look closely at a broccoli crown or Romanesco broccoli. If you cannot find one in the store, look up some pictures on the web. Can you see how the same patterns occur at different sizes?
Extra: Look around you. Can you find fractals in nature? Look at trees, leaves, ferns, and even clouds. Are these fractals too? Why or why not?
Extra: Some fractals consist of a figure containing infinitely many smaller and smaller versions of itself inside. Follow the link listed in the “More to explore” section to learn about the Sierpinski Triangle. Watch the video as it zooms in. Do you see how the pattern infinitely repeats itself at smaller and smaller scale? Can you draw a Sierpinski Triangle? Can you make up a fractal of this type yourself?
Extra: Look on the web for some images of computer-generated fractals and fractals occurring in nature. Do you find them artistic?
Observations and Results
Did you find recurring patterns at different scales? Did you see branching patterns occur? This is what is expected.
The paint likes to stick together. The interactions within the paint and between the paint and the plastic cover give rise to beautiful fractals when you first squeeze the paint between the covers and then release or pull the covers apart. Printing on glossy paper allows you to repeat the process and create patterns that show finer branching.
While the area of your figure is not changing much, its perimeter grows fast when more and more branching is added.
Fractals are common in nature. Trees, leaves, ferns, shells, lightning, and clouds are just some examples. Fractals are not only around us, but also inside us: our lungs and blood vessels show fractal characteristics. Fractal mathematics can help us describe and quantify these structures. Engineers also use fractals to create new products, like cell phone antennas. Fractal art is a form of digital art, where artists use computer-generated images to create intricate objects of beauty.
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Cleanup
- Wash all equipment with soapy water.
Additional Resources
- Earth's Most Stunning Natural Fractal Patterns, from Wired
- Teaching Fractals, from Ann Treacy's Doodles and Jots
