# Fractals

Fractals are forms which repeat themselves, changing scale by a constant factor. If we “zoom in” or “zoom out” the form should look the same. Its likely you will have to use your imagination to consider the very large or the very small aspects of a fractal.

## Models of Fractals

1) Hurricane Sandy

2) Sierpinski’s Triangle

3) Sierpinski’s Triangle Black and White

4) Fern Photo

5) Fibonacci Tree

### 1) Hurricane Sandy

NOAA took a photograph of Hurricane Sandy taken from a satellite.

At that moment in time a swirl could be identified. That swirl can be shown to comprise square units each with a quarter circle inscribed from corner to corner, and successively smaller, or larger by the Golden Ratio.

Obviously the hurricane has a maximum and minimum size, but it does show the Golden Ratio in sequence in a compelling way.

### 2) Sierpinski’s Triangle

This model uses the capability of the laser to create a perfect triangle, and then to create smaller triangles that can fit within.

We have used this large color model that is very nice for young children particularly. The protocol that has worked for us is 1) Invite them to play with the pieces and make a pattern or picture; 2) Invite them to try to “copy” your rule. You place one triangle face down in the center. This could go on a card that flips over with further details if a guide is not present; 3) Cheer on your guest as he or she continues the pattern; 4) Engage with your guests imagination to consider smaller and smaller triangles as well as bigger and bigger ones.

### 3) Sierpinski’s Triangle – Black and White

This model is made with reversible pieces to allow a variety of patterns.

You can have some other fractal patterns like this one,

and also make a clear case for for “fractal/not a fractal”.

Another nice connection here might be to include proofs of the sum of the colored areas. In Sierpinski’s the white areas are constantly and infinitely attacked by the color with non spared as the scale changes so these fractions should sum to 1. If we just color the left and right corner each time, we are constantly preserving 1/4 of the figure. The colored part is 1/2 + 1/2 (1/4) + 1/2 (1/16) + 1/2 (1/64)…. which can be shown to be equal to 2/3. The white part that is preserved is 1/4 + 1/16+1/64 +… which can be shown to be 1/3. So that’s fun.

### 4) Fern Photo

There are numerous examples of fractals in nature, but this one’s a classic and makes a clear case for self similarity with the frond composed of smaller fronds which are composed of smaller fronds. I don’t know if the scale holds constant in this example. A fern structure is also commonly used to generate fractal patterns.

### 5) The Fibonacci Tree

The rule for the tree is that a trunk splits and the new sprig cannot split till it grows a cycle into a branch. To think of this as a fractal, you have to think of the component parts across growth cycles. The basic building block can be replicated and joined to create the tree. Counting across the tree in comparable “generations” gives the fibonacci numbers,

but if you reframe it as one side of the split grows faster than the other, then you are back to a geometric growth of the form 2^n . A real tree example gives the most impact,

but we can also play with our etching of Fibonacci Tree to see this.