FRACTALS

WHAT ARE FRACTALS? Fractals are geometric figures, just like rectangles, circles, and squares, but fractals have special properties that those figures do not have. Fractals are shapes that consist of small parts that resemble the whole picture. In other words, when they are magnified, their parts are similar to the whole shape.

WHERE DID FRACTALS COME FROM? Most math you study in school is old knowledge. For example, the geometry you study about circles, squares, and triangles was organized around 300 B.C. But Fractals are considered relatively new mathematics. The first published information about fractals was in British cartographers encountered the problem in measuring the length of Britain coast. The coastline measured on a large scale map was approximately half the length of coastline measured on a detailed map. The closer they looked, the more detailed and longer the coastline became. They did not realize that they had discovered one of the main properties of fractals.

DIFFERENCE BETWEEN NON-FRACTAL SHAPES AND FRACTAL SHAPES a) As a non-fractal shape is magnified, no new features are revealed. Non-Fractals

b) The size of the smallest feature of a non-fractal shape is only one size.

a) As a fractal shape is magnified, ever finer new features are revealed. The shapes of the smaller features are kind-of-like the shapes of the larger features. Fractals

b) The size of the smallest features of a fractal shape has multiple ranges of size. There is no one specific size.

PROPERTIES OF FRACTALS There are two very important properties about fractals: 1. Self-Similarity If you look carefully at a fern leaf, you will notice that every little leaf (part of the bigger one) has the same shape as the whole fern leaf. You can say that the fern leaf is self-similar. The same is with fractals: you can magnify them many times and after every step you will see the same shape, which is characteristic of that particular fractal.

2. Fractional Dimensions We currently know the following dimensions: zero dimensional points, one dimensional lines and curves, two dimensional plane figures such as squares and circles, and three dimensional solids such as cubes and spheres. However, many natural phenomena are better described using a dimension between two whole numbers. So while a straight line has a dimension of one, a fractal curve will have a dimension between one and two, depending on how much space it takes up as it twists and curves.

HOW TO CREATE A FRACTAL 1. Take a familiar geometric figure and operate on it so that the new figure is more “complicated” in a special way. 2. Then in the same way, operate on that resulting figure, and get an even more complicated figure. 3. Do it again and again…and again. In fact, you have to think of doing it infinitely many times. **Each time you repeat this process, it is called one iteration.

FRACTALS IN NATURE Fractals can be found virtually everywhere in the natural world. Here are only a few examples. 1. Lightning2. Sea Shells

3. Snow Flakes4. Broccoli 5. Terrain from Satellite6. Pineapple

FAMOUS FRACTALS Here are a few famous fractals that we will be creating in this class. 1. Sierpinski’s Triangle2. Koch’s Snowflake

HOMEWORK Find a fractal in nature. Take a picture of the fractal and Ms. Robbins the picture. If you don’t have a digital camera, then try your best to hand draw the fractal. In the /drawing, explain what the fractal is and where you found it.