Fractals"Clouds are not spheres, mountains are not cones,

coastlines are not circles, and bark is not smooth, nor does lightning travel in a straight line."(Mandelbrot,

1983).

Fractals"Clouds are not spheres, mountains are not cones,

coastlines are not circles, and bark is not smooth, nor does lightning travel in a straight line."(Mandelbrot,

1983).

Definition: A fractal is an object or quantity that displays self-similarity, in a somewhat technical sense, on all scales.

- Wolfram MathWorld

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.

 

Dimension 0

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. 

Dimension 1

A plane has two dimensions - length and width, no depth.

It's an absolutely flat tabletop extending out both ways to infinity.

Dimension 2

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.

Dimension 3

Fractals can have fractional dimension. A fractal might have dimension of 1.6 or 2.4. How could that be? Let's investigate.

  This isn't a great picture of a fractal. It's really just an approximation of one. Fractals really are formed by infinitely many steps, not just the three of this one. So we have to remember that there are infinitely many smaller and smaller triangles inside the real fractal, and infinitely many holes (the black triangles) at the same time..

Fractional Dimension

In order to see how fractals could have dimension of a fraction, let's see what we mean by dimension in general. Take a self-similar figure like a line segment, and double its length. Doubling the length gives two copies of the original segment.

 

Self-similar

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.

Self-similar

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.

Self-similar

Do you see the pattern?

 Figure   Dimension   No. of Copies 

Line segment 1 2 = 2 1

Square 2 4 = 2 2

Cube 3 8 = 2 3

 Figure   Dimension   No. of Copies 

Line segment 1 2 = 2 1

Square 2 4 = 2 2

Cube 3 8 = 2 3

 Any Self-Similar Figure 

d n = 2 d

Formula for Dimension

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 not a part of the Sierpinski triangle

Sierpinski Triangle

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

Sierpinski Triangle

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

Fractional Dimension

 Figure   Dimension   No. of Copies Line segment 1 2 = 2 1 Sierpinsi's Triangle ? 3 = 2 ?

Square 2 4 = 2 2

Cube 3 8 = 2 3

 Any Self-Similar Figure  d n = 2 d

For the Sierpinski triangle consists of 3 self-similar pieces, each with magnification factor 2

How can we solve for d?

We begin with a straight line of length 1, called the initiator. We then remove the middle third of the line, and replace it with two lines that each have the same length (1/3) as the remaining lines on each side. This new form is called the generator, because it specifies a rule that is used to generate a new form.

Koch Curve

The rule says to take each line and replace it with four lines, each one-third the length of the original

You try to draw level 3.

Koch Curve Level 2

The rule says to take each line and replace it with four lines, each one-third the length of the original

Koch Curve Level 3

Koch Curve

We do this iteratively ... without end. The Koch Curve.

What is the length of the Koch curve?

Koch Snowflake

Koch Snowflake

What is the fractional Dimension of the Koch Curve?

What is the fractional Dimension of the Koch Curve?

Each line has become 4 self-similar copies with 3 for scaling factor!

D = log(N)/log(r) D = log(4)/log(3) = 1.26

Cantor Dust

Iteratively removing the middle third of an initiating straight line, as in the Koch curve, ... Initiator and Generator for constructing Cantor Dust. ...

this time without replacing the gap... Levels 2, 3, and 4 in the construction of Cantor Dust.

Calculating the dimension ... D = log(N)/log(r)

D = log(2)/log(3) = .63

We have an object with dimensionality less than one, between a point (dimensionality of zero and a line (dimensionality 1).

Cantor Dust

Sierpinski Carpet

The Sierpinski carpet is a plane fractal first described by Wacław Sierpiński in 1916. The carpet is a generalization of the Cantor set to two dimensions (another is Cantor dust)

Sierpinski Carpet

What is the fractional dimension?

Sierpinski Carpet

In the fractal, there are 8 identical figures, each of which has to be magnified 3 times to get the entire figure.

D= log 8 / log 3 which is approximately 1.89.

Sierpinski Carpet

Menger Sponge

Menger Sponge

It's fractal dimension equals log 20 / log 3, approximately 2.73

Menger Sponge