Did you notice that one of the songs in the movie Frozen mentioned fractals? My kids did!
So what is a fractal anyway?
A fractal is a shape that looks the same no matter how much you zoom in on it. It has self similarity: little pieces of it look exactly like the whole thing.
Think of a big stalk of broccoli. If you break off one piece, it looks like a small version of the original. Then you break off another piece, and it looks like the original. With broccoli, you eventually run out of pieces to break, but in a true mathematical fractal, you can keep looking at smaller pieces and they look exactly like the original.
Why do we care about fractals?
They are awesome. Okay, so maybe you want an answer that a non-math geek would appreciate. They can be used to create beautiful pictures, model shapes in the real world, and improve the technology in your cell phone.
Mandelbrot was the mathematician who first started serious work with fractals. His ideas were rejected at first, but eventually people realized that the real world is modeled better by fractals than by straight lines and simple curves.
You use fractals and see fractals everyday. Special effects in movies are often created by fractals. Why? Because the effects look more believable when they are detailed with this idea of self similarity than if they are smooth curves and straight lines. For example, the lava that shoots up as Obi-wan fights with Anakin Skywalker was CGI. It was created by drawing a curve and then using the computer to make curve be drawn with little versions of itself. This process was repeated a bunch of times, and the result was a very natural looking jet of lava. In other movies, fractals have been used to create entire landscapes. Fractals have greatly improved the movies that you know and love.
You use fractal technology every time you pick up your cell phone, because the antennas in cell phones are based on fractals. There is a great story about a short wave radio operator who attended one of Mandelbrot’s lectures. His landlord had always complained about his big antennas. After the lecture, he thought “what if an antenna was shaped like a fractal?” Then he bent a coat hanger into the shape of the Koch Curve (a famous fractal) and even though it was much smaller, it was the best antenna he ever had. Now that idea is used in all cell phones.
There are simple fractals like Fractal Trees, Sierpinski’s Triangle and the Koch Curve. I like to draw these when I am stuck in a boring meeting.
And there are beautifully complicated fractals that come from simple mathematical equations. Here is a video that shows the Mandelbrot Set. It is a beautiful fractal that is created from a surprisingly simple equation. This video zooms in on different places and you can see how the entire shape is embedded inside. The video starts slowly so you might want to jump to 1:30 to start.
Fractal Trees. Draw a segment for the trunk. Then make a “V” on it. Make more Vs on the new ends. Each time you draw, make the new segments smaller than the last “generation”. Have some fun with these: make different generations different colors, vary how wide the Vs are, and experiment with other ideas.
The Koch Curve. Basically you draw a line segment and then make an equilateral triangle in the middle of it. Then draw a triangle in the middle of each segment. Repeat.
Sierpinski’s Triangle. Draw andequilateral triangle. Connect the midpoints of the sides. Continue connecting the midpoints of the new triangles (but do not draw in the upside-down triangles).
While mathematicians discover these ideas, it was our Creator who put them in our universe. The more I study math, the more I am in awe of God. He gives us logic, math, beauty, and all of the other ideas we have in this life.