Fractal Dimension

Fractal Dimension of the Sierpinski Triangle

Let's use the formula for scaling to determine the dimension of the Sierpinski Triangle fractal. First, take a rough guess at what you might think the dimension will be. Less than 1? Between 1 and 2? Greater than 2? Since the Sierpinski Triangle fits in plane but doesn't fill it completely, its dimension should be less than 2. Let's see if this is true.

Start with the 0 order triangle in the figure above. The next iteration, order 1, is made up of 3 smaller triangles. And order 2 is made up of 9 triangles. So each iteration of the fractal has 3 times as many triangles, and N=3. Next we need to figure out the scaling factor, r. How much smaller is each triangle in order 1 than order 0? Look at the edge of each triangle in order 1, and you can see that the edge of each triangle is half the length of the edge of the triangle in order 0. So the scaling factor r=2.

That's all we need to know, and we can find the dimension by using the formula:

Use a calculator (or Google) to find the value for Log(3): [ ]

Find the value for Log(2): [ ]

Verify that log(3) / log(2) = 1.585

Note that dimension is indeed in between 1 and 2, and it is higher than the value for the Koch Curve. This makes sense, because the Sierpinski Triangle does a better job filling up a 2-Dimensional plane.

Next, we'll apply this same idea to some fractals that reside in the space between 2 and 3 dimensions.

Fractal Dimension of the Menger Sponge

How do the cubes scale from left to right? Starting with the cube on the left (order 0), we must figure out how many smaller cubes there are in each subsequent iteration. It's a little tricky to count the little cubes in the 1'st order cube, but the blue grid on the front face shows the scale of the smaller cubes. There are 8 cubes on the front face, 8 cubes on the back, and then 4 in between, making a total of 20 cubes, so N = 20.

Next, we need to figure out the magnification or scaling factor, r. The length of each of the 1'st order cubes is 1/3 of the length of the 0-order cube, so the magnification factor is 3. Using these values, we can compute the dimension as follows:

The dimension of the Menger Sponge is in between 2 and 3, which makes sense. It definitely is more than a 2-Dimensional object, but it does not completely fill up 3-Dimensional space either.

How many small cubes are there in the 2'nd order cube in the Menger Sponge? [ ]

How much samller are the cubes in the 2'nd order cube than the unit cube, i.e what fraction of the 0-order cube are they? [ ]