# A Fractal in a Fractal

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High School

### Type:

Mathematics, Computer Science

### Objective:

This experiment will explore whether fractals really repeat themselves infinitely.

### Research Questions:

• Are fractals really self-similar?
• How far must one zoom into a fractal to find an exact replica of the whole fractal?

### Introduction:

Have you ever stared into a fractal and wondered how many times it really repeats itself? This mathematical concept of a geometric, self-similar object called a fractal has been around for centuries, but now computers can give us a view of these complicated math objects and show them to a near infinite scale.

• Computer
• Printer
• Journal

### Experimental Procedure

1. Download a fractal-producing program and open up the Mandelbrot set fractal, a famous fractal.
2. To see the fractal extremely up close, set the maximum iterations very high, to about a few million. Record the level of magnification in your journal, and also note the area of the fractal you are exploring, such as “upper-left quadrant”.
3. Print the image out.
4. Next, find an even closer view of the fractal by zooming in on the computer. Use the printed image to hunt for an exact replica of the initial fractal. It might require zooming in and out several times.
5. If you can find a replica, print the image.
6. Zoom in once again, and find another replica of the fractal. Print any matches.
7. Go back to the original fractal, and attempt steps 4 to 7 again, working with a different area of the fractal.
8. Analyze your data. How long did it take you to find self-similarity? How much magnification did it take? Which areas on the fractal appear to be most self-similar?

Concepts: fractals, iterated equations, self-similarity