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15.3: Fractal Dimension

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In addition to visual self-similarity, fractals exhibit other interesting properties. For example, notice that each step of the Sierpinski gasket iteration removes one quarter of the remaining area. If this process is continued indefinitely, we would end up essentially removing all the area, meaning we started with a 2-dimensional area, and somehow end up with something less than that, but seemingly more than just a 1-dimensional line.

To explore this idea, we need to discuss dimension. Something like a line is 1-dimensional; it only has length. Any curve is 1-dimensional. Things like boxes and circles are 2-dimensional, since they have length and width, describing an area. Objects like boxes and cylinders have length, width, and height, describing a volume, and are 3-dimensional.

fr14.svg

Certain rules apply for scaling objects, related to their dimension.

If I had a line with length 1, and wanted scale its length by 2, I would need two copies of the original line. If I had a line of length 1, and wanted to scale its length by 3, I would need three copies of the original.

fr15.svg

If I had a rectangle with length 2 and height 1, and wanted to scale its length and width by 2, I would need four copies of the original rectangle. If I wanted to scale the length and width by 3, I would need nine copies of the original rectangle.

fr16.svg

If I had a cubical box with sides of length 1, and wanted to scale its length, its width, and its height by 2, I would need eight copies of the original cube. If I wanted to scale the length, width, and height by 3, I would need 27 copies of the original cube.

fr17.svg

Notice that in the 1-dimensional case, copies needed = scale.

  • In the 2-dimensional case, copies needed = scale2.
  • In the 3-dimensional case, copies needed = scale3.

From these examples, we might infer a pattern.

Scaling-Dimension Relation

To scale a D-dimensional shape by a scaling factor S, the number of copies C of the original shape needed will be given by:

Copies = Scale Dimension, or C=S

Example 5

Use the scaling-dimension relation to determine the dimension of the Sierpinski gasket.

Solution

Suppose we define the original gasket to have side length 1. The larger gasket shown is twice as wide and twice as tall, so has been scaled by a factor of 2.

fr18.svg

Notice that to construct the larger gasket, 3 copies of the original gasket are needed.

Using the scaling-dimension relation C=SD, we obtain the equation 3=2D

since 21=2 and 22=4, we can immediately see that D is somewhere between 1 and 2; the gasket is more than a 1 -dimensional shape, but we've taken away so much area its now less than 2-dimensional.

Solving the equation 3=2D requires logarithms. If you studied logarithms earlier, you may recall how to solve this equation (if not, just skip to the box below and use that formula):

3=2DTake the logarithm of both sideslog(3)=log(2D)Use the exponent property of logslog(3)=Dlog(2)Divide by log(2)D=log(3)log(2)1.585The dimension of the gasket is about 1.585

Scaling-Dimension Relation, to find Dimension

To find the dimension D of a fractal, determine the scaling factor S and the number of copies C of the original shape needed, then use the formula

D=log(C)log(S)

Try it Now 2

Determine the fractal dimension of the fractal produced using the initiator and generator

fr19.svg

Answer

fr21.svg

Scaling the fractal by a factor of 3 requires 5 copies of the original. D=log(5)log(3)1.465

We will now turn our attention to another type of fractal, defined by a different type of recursion. To understand this type, we are first going to need to discuss complex numbers.


This page titled 15.3: Fractal Dimension is shared under a CC BY-SA 3.0 license and was authored, remixed, and/or curated by David Lippman (The OpenTextBookStore) via source content that was edited to the style and standards of the LibreTexts platform.

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