Mesmerizing Fractals

Key concepts
Mathematics
Geometry
Fractals
Scale

Introduction
Do you ever wonder what mathematicians study—and why? Most of what they do is complex and difficult to understand, but fractal art might give us a glimpse. Mathematicians study fractals, which are naturally occurring figures used in many branches of science and technology. But you don’t have to be a mathematician to appreciate their beauty. Did you know you can also create one? In this activity you will get to take out some paint to make artwork—and discover how common fractals are.

Background
Fractals are geometric figures. They are difficult 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 such as 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 three or five or any other number that is not a whole power of two. This is characteristic for fractals. They have a fractional dimension, such as 8/5.

A final surprising fact: some fractals can show an infinite perimeter, even though 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 (Finger paint paper works best.)
• Magnifying glass (optional)

Preparation

• Protect your workspace.
• Disassemble the CD jewel case so you have two loose covers.
• Place one of the covers on your workspace with the flat part facing up, with the edges of the cover below it.

Procedure

• 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, flat part facing down, 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. As you do, try not to slide the covers over each other, rather 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. A pattern that reappears 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 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 internet. 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 internet for some images of computer-generated fractals and fractals occurring in nature. Are there some that you find particularly impressive?

Observations and results
Did you find recurring patterns at different scales? Did you see branching patterns occur?

The paint likes to stick together. The interactions within the paint and between the paint and the plastic cover give rise to impressive 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.

Although 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 they are 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 such as cell phone antennas. Fractal art is a form of digital art in which artists use computer-generated images to create intricate objects.

Cleanup
Wash all equipment with soapy water.

More to explore
What Are Fractals?, from the Fractal Foundation
Earth's Most Stunning Natural Fractal Patterns, from Wired
Sierpinski Triangle, from Rice University

This activity brought to you in partnership with Science Buddies