1. "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).

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

3. 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.

4. 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.

5. A plane has two dimensions - length and width, no depth. It's an absolutely flat tabletop extending out both ways to infinity.

6. 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.

7. 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..

8. 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

9. 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.

10. 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.

11. Figure Dimension No. of Copies Line segment12 = 2 1 Square24 = 2 2 Cube38 = 2 3

12. Figure Dimension No. of Copies Line segment12 = 2 1 Square24 = 2 2 Cube38 = 2 3 Any Self- Similar Figure dn = 2 d

13.  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

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

15.  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. Figure Dimension No. of Copies Line segment12 = 2 1 Sierpinsi's Triangle ?3 = 2 ? Square24 = 2 2 Cube38 = 2 3 Any Self- Similar Figure dn = 2 d

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

17.  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.

18.  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.

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

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

21. What is the length of the Koch curve?

25. 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

26. 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.

27.  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).

29.  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)fractalWacław SierpińskiCantor setCantor dust

30.  What is the fractional dimension?

31.  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.

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