# Fractals

### Suzanne Alejandre

Magic Squares || Multicultural Math Fair || Polyhedra || Tessellations

### What is a fractal?

#### From the Fractal FAQ:

"A fractal is a rough or fragmented geometric shape that can be subdivided in parts, each of which is (at least approximately) a reduced-size copy of the whole. Fractals are generally self-similar and independent of scale."

"There are many mathematical structures that are fractals; e.g. Sierpinski triangle, Koch snowflake, Peano curve, Mandelbrot set, and Lorenz attractor. Fractals also describe many real-world objects, such as clouds, mountains, turbulence, and coastlines, that do not correspond to simple geometric shapes."
The Fractal FAQ was created and edited by Ken Shirriff through September 26, 1994. The current editor is Ermel Stepp.

#### Alan Beck writes:

"Basically, a fractal is any pattern that reveals greater complexity as it is enlarged. Thus, fractals graphically portray the notion of 'worlds within worlds' which has obsessed Western culture from its tenth-century beginnings."

Beck further explains that when we look very closely at patterns that are Euclidean, the shapes look more and more like straight lines, but that when you look at a fractal up close you see more and more details.

Beck continues, "Whether generated by computers or natural process, all fractals are spun from what scientists call a 'positive feedback loop'. Something - data or matter - goes in one 'end', undergoes a given, often very slight, modification, and comes out the other. Fractals are produced when the output is fed back into the system as input again and again."

Alan Beck: What Is a Fractal? And who is this guy Mandelbrot?

### How can I find out more about fractals?

There's a lot more on the Web about fractals. As you read, you may see why mathematicians find fractals so interesting. Look at the Koch curve (sometimes called the Koch snowflake): as you keep dividing and dividing and dividing this shape....the perimeter keeps getting infinitely larger, yet the area is finite!

The following Web pages will provide you with additional information. Remember to keep in mind the vocabulary we have been using as you view them.

As you browse, remember to take notes to add to your journal entries.