PS 3218 1.16 Fractals Properties (Part 1)

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.

What are the properties of fractals?

1. Self-Similarity. If parts of a figure contain small replicas of the whole, then the figure is called self-similar. If the figure can be decomposed into parts which are exact replicas of the whole, then the figure is called strictly self-similar structure contains an exact replica of the whole.

2. Fractal Dimension

Table of Contents Introduction Why study fractals? What's so hot about fractals, anyway? Making fractals Sierpinski Triangle Using Java Math questions Sierpinski Meets Pascal Jurassic Park Fractal Using JAVA It grows complex Real first iteration Encoding the fractal World's Largest Koch Snowflake Using Java Infinite perimeter Finite area Anti-Snowflake Using Java Fractal Properties Self-similarity Fractional dimension Formation by iteration For Teachers Teachers' Notes Teacher-to-Teacher Comments My fractals mail Send fractals mail Fractals on the Web The Math Forum Other Math Lessons by Cynthia Lanius Awards This Site has received

A point has no dimensions - no length, no width, no height. That dot is obviously way too big to really represent a point. But we'll live with it, if we all just agree what a point really is. A line has one dimension - length. It has no width and no height, but infinite length. Again, this model of a line is really not very good, but until we learn how to draw a line with 0 width and infinite length, it'll have to do. A plane has two dimensions - length and width, no depth. It's an absolutely flat tabletop extending out both ways to infinity. Space, a huge empty box, has three dimensions, length, width, and depth, extending to infinity in all three directions. Obviously this isn't a good representation of 3-D. Besides its size, it's just a hexagon drawn to fool you into thinking it's a box. Fractals can have fractional (or fractal) dimension. A fractal might have dimension of 1.6 or 2.4. How could that be? Let's investigate below.

Just as the images above weren't very good pictures of a point, line, plane, or space, the drawing meant to be the Sierpinski Triangle has limitations. Remember as we continue that fractals are really formed by infinitely many steps. So there are infinitely many smaller and smaller triangles inside the figure, and infinitely many holes (the black triangles).

Let's look further at what we mean by dimension. Take a self-similar figure like a line segment, and double its length. Doubling the length gives two copies of the original segment.
Take another self-similar figure, this time a square 1 unit by 1 unit. Now multiply the length and width by 2. How many copies of the original size square do you get? Doubling the sides gives four copies.
Take a 1 by 1 by 1 cube and double its length, width, and height. How many copies of the original size cube do you get? Doubling the side gives eight copies.

Let's organize our information into a table.

Figure	Dimension	No. of Copies
Line segment	1	2 = 21
Square	2	4 = 22
Cube	3	8 = 23

Do you see a pattern? It appears that the dimension is the exponent - and it is! So when we double the sides and get a similar figure, we write the number of copies as a power of 2 and the exponent will be the dimension.

Let's add that as a row to the table.

Figure	Dimension	No. of Copies
Line Segment	1	2 = 21
Square	2	4 = 22
Cube	3	8 = 23
Doubling Similarity	d	n = 2d

We can use this to figure out the dimension of the Sierpinski Triangle because when you double the length of the sides, you get another Sierpinski Triangle similar to the first.

Start with a Sierpinski triangle of 1-inch sides. Double the length of the sides. Now how many copies of the original triangle do you have? Remember that the black triangles are holes, so we can't count them.

Doubling the sides gives us three copies, so 3 = 2^d, where d = the dimension.

But wait, 2 = 2¹, and 4 = 2², so what number could this be? It has to be somewhere between 1 and 2, right? Let's add this to our table.

Figure	Dimension	No. of Copies
Line Segment	1	2 = 21
Sierpinski's Triangle	?	3 = 2?
Square	2	4 = 22
Cube	3	8 = 23
Doubling Similarity	d	n = 2d

So the dimension of Sierpinski's Triangle is between 1 and 2. Do you think you could find a better answer? Use a calculator with an exponent key (the key usually looks like this ^ ). Use 2 as a base and experiment with different exponents between 1 and 2 to see how close you can come. For example, try 1.1. Type 2^1.1 and you get 2.143547. I'll bet you can get closer to 3 than that. Try 2^1.2 and you get 2.2974. That's closer to 3, but you can do better.

3. Iterative Formation

Fractals are often formed by what is called an iterative process. Here's what I mean. To make a fractal: Take a familiar geometric figure (a triangle or line segment, for example) and operate on it so that the new figure is more "complicated" in a special way. Then in the same way, operate on that resulting figure, and get an even more complicated figure. Now operate on that resulting figure in the same way and get an even more complicated figure. Do it again and again...and again. In fact, you have to think of doing it infinitely many times. You can observe this iterative process in all the fractals that we make in this unit: Sierpinski's Triangle Koch Snowflake Anti-Koch Snowflake The Jurassic Park fractal Not every iterative process produces a fractal. Take a line segment and chop off the end. What is the resulting figure? Just another line segment - not "complicated" at all, and not a fractal. You could continue the iterative process over and over, chopping off the end of the line segment, but it would just become a shorter and shorter line segment, not "complicated", not fractal. Below is a picture of a similar iterative operation that is fractal. Take a line segment (see below) and remove the middle third. What is the resulting figure? Hmmm. That's a more complicated figure. It's a line segment with a hole in it. Repeat the process on that figure. In other words, remove the middle third of both of those sections. This produces an even more complicated figure. Now think of doing this infinitely many times. In fact, this is a famous fractal called Cantor Dust.