Ever notice the similarities between lightning bolts and river deltas? These two things are examples of a pattern in nature called a “Fractal.” In this activity, you will get a striking hands-on introduction to fractal patterns, and see one form right in front of your eyes! Come along on a race, er, bet?, between Bee and Treebeard as they attempt to determine the *correct* length of the coastline between Camp Magruder and Camp Westwind! 

Hmmm...That is a crazy conundrum! How in the world did Bee run 5 more miles than Treebeard?! Write your hypothesis in your nature journal.

At the beginning of this lesson we said that these fractal patterns are reminiscent of  lightning bolts and river deltas.  Make a list in your field journal of at least four other things these fractal patterns remind you of, or where else you’ve seen something like this.  

Can you identify what the photo below is? Write your guess in your field journal. 

This is just another image of a river delta (like the one on the cover of this lesson next to the lightning bolt)! This pattern is typically formed when water flows from tributary streams into a central river, or flows out from a main river into several branches.

Now can you identify this object?

Imagine the Jeopardy theme song playing

It's sunflower seeds!

As the fractal patterns formed in nature repeat themselves at different scales, each smaller section of the whole is similar to the large scale whole, but never an exact copy. For example, if you break a floret of cauliflower or broccoli off the larger head, you can see that it's like a miniature version of that larger head, but it's not an exact replica. These repeating but nonidentical patterns are called “self-similar”.  Now, watch 2-3 minutes of the Mandelbrot Zoom videos on Youtube, then come back and keep reading!

 

Perhaps you noticed how those swirls were all very similar, and certainly satisfying to look at (like a mandala!), but they’re not perfectly or precisely identical. 

Even though nature can’t generate a perfectly precise 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 completely identical to the whole object. In this kind of fractal you can’t tell the difference between the whole object and a magnification of a smaller section! The next image shows a mathematical fractal, the “Sierpinski Triangle”.

 

Alright, let’s see what Bee and Treebeard have learned about fractals! Watch the video below. 

 

  

Spend 5-10 minutes answering this question in your field journal: 
Explain in your own words why neither Bee nor Treebeard won the bet.

 

Now, try creating your own fractal art project like Bee and Treebeard did in the video!
First, gather your materials: 

• A straightened paper clip, toothpick, or bamboo skewer (any straight line object that is okay to get paint on)
• Some type of liquid paint in any color you like! Good options include: water-based acrylic gloss enamel model paint, acrylic paint, poster paint, latex paint...etc. Get creative and try a few different options to see how that affects your experiment! Test out using any liquid-based object: frosting, margarine, thick jam...
• Two pieces of clear plastic or glass of the same size, the smoother and more rigid the better, get creative with the materials you have at home!
  • Examples: CD cases, microscope slides, #6 plastic like a deli container that can be cut with scissors, or clear acrylic plastic; the pieces of plastic don’t have to be square, but it kind of helps.
  •  As small as 1” x 1” to as large as 6” x 6”
• Optional: transparent packing tape

Written Instructions:

Use a straightened-out paper clip to stir the paint. Then use it to place a tiny drop of paint (a
little goes a loooong way folks!) at the center of one of the plastic pieces, which are your
“Plates.”

Place the second “plate” on top of the first, covering the paint at an askew angle. Be sure not to line up the edges of the top plate 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 the next photo.

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. This should form a fractal pattern! What do you notice?

Once the plates are separated, observe the patterns on each one, and write down your observations in your field journal. 

• Did you notice that the patterns are mirror images of each other?
• What do you think the patterns look like? 
• Do they remind you of anything? 

If you want to preserve your fractal pattern as a work of art, let the paint patterns dry. If you want to try again with different liquid variables, simply wash your plates and start over!

Alternative Construction for art-loving masters:
Cover your plates 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.

What's Going On?
Your fractals are the result of a process called “viscous fingering”.  “Viscous” means “having a thick, liquid consistency.” So as the viscous paint is squeezed between the plates, it spreads out evenly in all directions into the less viscous air layer. This creates a stable, disk-shaped boundary between paint and air. (Think about when you overfill a glass of water juuust a little bit so that the surface tension of the water is so strong it makes that bubble over the glass and doesn’t spill...That’s a stable bubble-shaped boundary between water and air!) 

When the plates are pulled apart, that pulling force enables the less viscous air to push into the more viscous paint, creating an unstable boundary. (Ever heard these words: “every action has an equal and opposite reaction”? Pulling force→ pushing air!)

These 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. This “viscous fingering” process happens super quick!! It might be hard to notice, but the result is what you’re seeing in your paint. 

We know this all sounds kinda complicated, and it’s okay if that didn’t totally make sense to you.  Read those paragraphs once more, then do the experiment again--paying close attention to the paint as you pull your plastic pieces apart.  

Are you still unconvinced with this whole coastline theory thing? Well you’re in luck, because next, we’re going to recreate the coastline experiment Bee and Treebeard ran (pun intended!) from our very own homes! (Nice, no sand in your shoes!) 

Watch the video below to get an idea of the project:

 

Watch this next video to learn how to tie a slip knot

 

Alright, now gather the materials: 

• Some cardboard (thick cardboard like from an amazon box rather than thin cereal box cardboard.  Styrofoam will work too, just something you can push thumbtacks into!)
• 10-20 thumbtacks
• Coloring materials
• 2 different strings (different colors, markings, textures, or some way to tell them apart)
• A ruler

On your cardboard, draw your very own coastline! It can be as wiggly or as straight as you want, but be sure to provide at least a few wiggles (to reflect reality!) or else this experiment won’t really work.  The more wiggles, the more obvious the results will be.  

Now,  push 1 thumbtack on the furthest left edge and 1 thumbtack on the furthest right edge of your coastline.  These will be the start and end points for your measurement.  Then pin a few more points between those two along your coastline.  Use only half of the thumbtacks  you’ve gathered.  Then, take one of your strings and wind it end to end on your coastline, being sure to wrap a loop on each thumbtack.  Tie a knot at the end or cut it exactly so you have a precise measurement of your coastline.  

Next, add in the rest of your thumbtacks, being sure to get really detailed and all up in those nooks and crannies of your coastline!  Now take your second string and wrap it end to end again, and cut it at the end. 

Finally, line up your two strings next to each other.  Using your ruler, record each strings’ length.  What do you notice? Write down these observations in your nature journal.  

So What?
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. 

Final Reflection: 
Go to the website Google Earth Fractals written by Paul Bourke  and explore a bit.  Pick a favorite fractal.  In your field journal, write a poem or short story, or draw an image inspired by it.

To end our lesson, spend 10 or so minutes in your field journal reflecting on this question: Which country in the world has the longest coastline? (Is this an answerable question? Why or why not?) 

Going Further: Can’t get enough of these fractals? Read below to keep exploring! 

Algorithms 
The Sierpinski Triangle and the Cantor Set are two classic fractals whose algorithms can be easily understood. Look these names up in google to get a nice idea of how a mathematical fractal is generated. 

Benoit Mandelbrot 
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!