Lightning bolts, river deltas, tree branches, and coastlines are all examples of patterns in nature called fractals. In this Snack, you get a striking hands-on introduction to fractal patterns and how they’re formed.
- Paper clip, toothpick, bamboo skewer, or nail
- Solvent-based, gloss enamel model paint, assorted colors. Water-based, acrylic gloss enamel model paint works almost as well, is nontoxic, and cleans up with water but takes longer to dry. You can also try acrylic paint, poster paint, frosting, frosting gels, margarine, latex paint, or other materials.
- Two same-sized pieces of clear plastic or glass, the smoother and more rigid the better, 1 in x 1 in (2.5 cm x 2.5 cm) to as large as 6 in x 6 in (15 cm x 15 cm). Examples: CD jewel cases, microscope slides, #6 plastic used for deli containers that can be cut with scissors, or clear acrylic plastic; the pieces of plastic don’t have to be square.
- Optional: transparent packing tape
Use a straightened-out paper clip, a toothpick, a bamboo skewer, or a nail to stir the paint. Then use the same implement to place a tiny drop of paint at the center of one of the plastic pieces, which are your plates (click to enlarge photo below).
Place the second plate on top of the paint, but don't line up the edges of the top plate exactly with the edges of the bottom plate. (You will be pulling the plates apart, and if the edges are lined up this may be difficult; see photo below.)
Squeeze the two plates together firmly, so that the paint drop forms the thinnest possible circular layer between them. Notice the paint spreads into a disk (see photo below).
Carefully pull the plates apart as shown below. Do not slide them apart. It's very important that you pull the plates straight off one another. Watch air flow into the paint as you pull the plates apart, forming a fractal pattern.
Once the plates are separated, observe the patterns on each one. Notice that the patterns are mirror images of each other. What do you think the patterns look like? Do they remind you of anything? Let the paint patterns dry on the plates if you want to preserve them.
Cover your pieces of plastic with transparent packing tape so you can peel off the fractal pattern when the paint is dry. This allows you to reuse the plastic, or to tape the pattern to a piece of paper or use it in some other creative way. The tape has to be wide enough to allow the pattern to be created on a single piece; two-inch-wide transparent packaging tape works well.
Your fractals are the result of a process called viscous fingering: As the paint is squeezed between the plates, the viscous paint spreads out evenly in all directions into the less viscous air layer, creating a stable, disk-shaped boundary.
When the plates are pulled apart, the less viscous air penetrates the more viscous paint, creating an unstable boundary. Small indentations of air grow and become fingers of air. Random indentations in these fingers grow as well. By the time the two plates are separated, the fingers of air have formed intricate branching structures in the paint.
The patterns created with this process often remind people of tree branches or root systems, river deltas, or lightning bolts, all of which are outstanding examples of fractal patterns in nature. A few others are clouds, coastlines, jellyfish tendrils, coral reefs, and blood vessels in the lungs. The photo below shows another example of a natural fractal pattern, the branching of a river. This is typical of the pattern formed when fluid flows from tributaries into a central stream or flows out from a main course into several branches.
All the fractal patterns formed in nature—including the ones you just made—are generated by random processes. As the patterns repeat themselves at different scales, each section of the whole is similar to large- and smaller-scale structures, but are never an exact copy. If you break a floret of cauliflower or broccoli off the larger head, for example, you can see that it's like a miniature version of the larger head, but it's not an exact replica. These repeating but nonidentical patterns are called self-similar.
Even though nature can’t generate a perfect fractal pattern, mathematicians can. The fractals they create are called perfect or mathematical fractals. If you look at a small section of a mathematical fractal, the section will be identical to the whole object. And if you were to select an even smaller piece from the first section and magnify it, this piece would also duplicate the whole. In fact, in this kind of fractal you can’t tell the difference between the whole object and a magnification of any section you select—no matter how small! The image below shows a classic mathematical fractal, the Sierpinski Triangle, or Sierpinski Gasket.
Make a fractal pattern on clear tape (see Alternative Construction above). Stick the resulting two pieces of tape to a piece of paper, and create art or poetry inspired by the fractal images.
The Sierpinski Triangle and the Cantor Set, or Cantor Dust, are two classic fractals whose algorithms (set of instructions for their construction) can be easily understood. Look these up to get a nice idea of how a mathematical fractal is generated.
The word fractal was first used in 1975 by the Polish-born mathematician Benoit Mandelbrot (1924–2010). The word derives from the Latin frangere (to break) and fractus (broken, uneven). Mandelbrot spent much of his working life in both France and the United States and is credited with the development of fractal geometry. His various studies included stock-market fluctuations, the turbulent motion of fluids, and the distribution of galaxies.
Fractals helps us to understand many different areas of science, including crystal growth, earthquake processes, meteorology, and polymer structure, to name just a few. Fractals are particularly significant in the field of chaos theory, which seeks to explain apparently random behavior that occurs within a system.
This Snack was developed by the educational staff at the Boston University Center for Polymer Studies.
Barnsley, Michael, Fractals Everywhere. San Diego, Academic Press, 1988.
Mandelbrot, Benoit, The Fractal Nature of Geometry. San Francisco: W. H. Freeman, 1982.
Peitgen, Heinz-Otto and Peter Richter, The Beauty of Fractals: Images of Complex Dynamical Systems. Berlin: Springer-Verlag, 1986.