Category Archives: Fractals

Benoit Mandelbrot

Mandelbrot

Benoit Mandelbrot and Mandelbrot Set

Today (November 20) is the birthday of Benoit Mandelbrot (1924-2010). Mandelbrot is known for his work in fractal geometry, specifically hIs “theory of roughness” and his discovery of the Mandelbrot set. Mandelbrot also coined the term “fractal”.

Mandelbrot was born in Poland, but his family fled to France to escape Nazi persecution. He studied mathematics at university in Paris, and then came to the U.S. to pursue a master’s degree in aeronatics at CalTech.  Mandelbrot returned to Europe for a time, but he then took a job at IBM in New York, where he worked for 35 years.

Other mathematicians had studied the mathematical sets we now know as fractals, but Mandelbrot had the advantage of access to IBM computers. Fractal sets are created by feeding a value into an equation, and then taking that result and feeding it back into the equation.  Up until the advent of computers, calculations had to be done by hand, limiting the number of iterations that were possible. Using computers, Mandelbrot was the first to do millions of iterations, creating his now famous Mandelbrot set.

Nova did a great show on fractals and Mandelbrot, which you may have seen in a previous post. But, if you missed it, hear it is again.

Fractal Education: The Dragon Curve

I recently wrote about using the Sierpinski triangle to teach students about fractals. Another relatively easy fractal for young students to make is the dragon curve fractal. The dragon curve is created by folding a strip of paper in half over and over (in the same direction) and then placing all folds at 90o. The pictures below show what the curve looks like after 1, 2, 3, and 4 folds. (This is a great time to introduce the term ‘iteration’ to students. Iteration means the process of repeating. The curve with 4 folds is the 4th iteration.)

DragonCurve1-4Folds

If you start with a really long piece of paper, you may be able to fold it 5 or 6 times, but that is about the limit. If you could fold it 14 times, it would look like the picture below. If you kept folding, the curve would have the same shape, it would just be folded in on itself even more. Notice how each “section” of the curve looks like a smaller version of the whole curve.

dragon_curve

It can be difficult to make all the folds to lie at 90o angles, even with a small number folds, so I like to have students tape their curve down onto a piece of paper. If you size things correctly, you can put your folded paper dragon curve onto a picture of a high-iteration dragon curve, as shown below – a great example of how a fractal looks similar at different scales.  You can print my directions and handout here.

IMG_0519

You can also use colored paper or cardstock to make some dragon fractal art. Take a look at these lovely dragon curves at CutOutFoldUp.com. If you are interested in the math behind the dragon curve, this page at The Fractal Umbrella explains how to mathematically describing the fold sequences.

Numberphile has a great video about the dragon curve (below) that covers many fascinating facts about the curve, including why it is also known as the Jurassic Park fractal.

Fractal Branching

One of the most common fractal patterns found in nature is branching. Fractal branching is seen in tree branches and leaf veins, lightning, river deltas, mountaintops, blood vessels and bronchi in the lungs, and many other places. Can you think of any examples?

FractalBranching

Fractal Education: Sierpinski Triangle

A great activity for teaching kids about fractals is making Sierpinski triangles – one of the easiest fractals to draw.

The Sierpinski triangle is formed using this process:

  1. Take a point-side-up triangle
  2. Connect midpoints of the three sides to create a point-side-down triangle.
  3. Repeat process with the resulting point-side-up triangles.
  4. Repeat as many times as you want.

This example shows taking the process through 4 iterations.

680px-Sierpinski_triangle_evolution.svg

You can print out the starting triangle from:

Fractal Foundation: Sierpinski Triangle (includes worksheet and instructions)

Spitz: Fractal Pack 1: Educators’ Guide (includes Sierpinski triangle template and other information for teaching about fractals)

If you have a classroom of kids, you can take their Sierpinski triangles and put them together to make a larger triangle – you just need a power of 3 to make a complete Sierpinski triangle (3, 9, 27, etc.).

photo-6

Also, you can blow their minds by showing this Sierpinski Zoom video to show that the triangles can go on forever!

NOVA Exposes the Hidden Dimension of Fractals

Mandelbrot Set created by Wolfgang Beyer with the program Ultra Fractal 3.

Mandelbrot Set created by Wolfgang Beyer with the program Ultra Fractal 3.

Whether or not you’ve heard of fractals before, the NOVA documentary, Hunting the Hidden Dimension, will amaze you with how cool they really are. A fractal is a geometric pattern that is repeated at smaller and smaller scales, producing shapes that can’t be represented by classical geometry. When mathematicians first started toying with the idea of fractals, they seemed so strange and foreign they were known as “monsters”. Now we see that they aren’t so foreign. In fact, they are everywhere – the branching of trees and blood vessels in our bodies, coastlines, clouds. Isn’t it amazing how even the strangest mathematical concepts seem to lead back to the natural world?

Hunting the Hidden Dimension is a fascinating look at fractals, covering the history of their study, from the 19th century, when they were known as “monsters”, to current applications, such as CGI and cell phone antennae. We also learn about the life and work of Benoit Mandelbrot, the man who developed fractal geometry as a field of mathematics and coined the term “fractal”, his advantage being that he came along at a time when computers were becoming available to tackle such problems. The “hidden dimension” in the title refers to the “fractal dimension”. You’ve heard of things being 2-dimensional or 3-dimensional, but fractal geometry can describe shapes with non-integer dimensions like 1.3 or 2.6.

I watched this documentary with my kids and I have shown clips of it to my after-school math group. The kids especially like seeing the beautiful images of the Mandelbrot set and seeing how fractals were used in the making of the latest Star Wars movies.

Hunting the Hidden Dimension is available for free on Hulu and is currently available on YouTube (embedded below).