COMPLEXITY

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Fractal Dimensions and Complexity

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Complexity, You Say?

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The many types of fractal analysis that exist rest on some type of fractal dimension (DF). There are many types of DF, but all can be condensed into one category—they are meters of complexity.

The word "complexity" is part of our everyday lives, of course, but fractal analysts have kidnapped it for their own purposes in fractal analysis. You know the concept already intuitively - you see it with magnification and you ponder it when you discuss image resolution. In fractal analysis, complexity means:

A change in detail with change in scale.

Something you may not already be familiar with is that complexity can be measured by fractal analysis.

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Fractal Dimensions

A fractal dimension (DF) is a measure of complexity. In practice, we infer a DF for a pattern from measurements we assume correspond to change in detail with change in scale. The measurements FracLac uses come from Box Counting, which is a way of sampling an image. In addition to finding complexity or rates of change in detail with scale, box counting measures heterogeneity as a quantity known as lacunarity.

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Scaling Rules

A DF is, in essence, a scaling rule comparing how a pattern's detail changes with the scale at which it is considered - this is what we mean by complexity.

In general, we deduce the value of the scaling rule, the DF, from knowing how something scales. Scaling something means looking at it at a finer and finer level of resolution, Think of using a magnifying glass through to a microscope to inspect something and seeing how it changes. Formally, the scaling rule idea is about the relationship between N, the number of pieces we find at each new resolution and ε , the scale used to get the new pieces. We say that N is proportional to the inverse of epsilon raised to some exponent:

Eq. 1) N ∝ ε^-D_F

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At this point, you may be thinking, "N?? I don't need no stinking N". You may know with great certainty that whenever you have changed the magnification at which you were viewing something, the number of pieces always stayed the same and that only the size of the image changed. But ask yourself this - did the detail change? Detail is what we mean when we say the number of pieces .

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Quantifying Scaling

To understand how all this fits together in the calculations for scaling rules and DFs, let's look at things differently than everyday life usually asks us to. First consider something you know, patterns such as the familiar Euclidean shapes of elementary geometry.

Simple Scaling

One shape for which scaling is easy to grasp is an ordinary line. A line, when scaled by, as an example, 1/3, can be seen to be made up of 3 pieces, each 1/3 the length of the original. Nothing cosmic to that. I won't even draw a diagram because it will surely just bore you. But it is kind of interesting to know that it gives us a use for the N in the scaling relationship, and we can figure out that DF = 1.00 in this situation just by substituting into the equation thus

3 = (1/3)^-1

Maybe you have already expanded this idea and figured out that all this is to say, too, that when scaling a filled square by 1/2, there will always be 4 new pieces, each 1/4 the area of the original, and D would be equal to 2:

4 = (1/2)^-2

Not So Simple Scaling

This may seem trivial—that the dimension (or complexity) of a line is 1 and of a filled square 2—but the decidedly untrivial part is that this sort of scaling, the scaling we know and love, is not necessarily the only kind of scaling possible. The Koch fractal line illustrated here, for example, scales into 4 new pieces each 1/3 the length of the original. This phenomenon is the crux of the matter.

You can see how the pattern can be formed in the animation. Basically, what happens is that the starting piece is scaled down to 1/3 the length it was, then that piece is laid down four times to make a new one that is, end to end, the length of the original, but has more pieces. This goes on forever and the infinite result, alas, we mere mortals cannot see, but is, nonetheless, the Koch fractal pattern.

That is, in the final pattern (not the iterations shown here but just the last version), you can imagine using a magnifying glass, then an optical microscope, then an electron microscope, to look at a smaller and smaller chunk of the original, where you will always see the exact same pattern, never the smaller component we see in the iterative diagram. To repeat this key concept emphatically, for every 1/3 scaling there are not 3 but 4 new pieces. The consequence is that you will never see the change in relative segment size that the iterative diagram shows—rather, as illustrated in the figure below, you will always see the exact same relationship when you scale down. It will make you dizzy.

The definitely untrivial point here is that in contrast to the line and square considered above, the scaling rule, DF, for this pattern, even if we could perceive it's infinite nature, is not so obvious—the numbers are

4 = (1/3)^-DF

and this we cannot solve by simple substitution into the scaling rule.

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How is a Fractal Dimension Calculated?

Logs and Limits

For the Koch pattern, you have to get out your math assistance device of choice and calculate the DF. I calculate a DF for any case like this by solving the general equation for the scaling rule:

N = Aε^-DF

for its variable, DF. To do this, I use algebra and logs. The definition of a logarithm is that it is the exponent to which we raise a base to get a result. So, we use that definition to rearrange the equation, as below. Then we use the rule that the logarithm itself (DF) is also equal to the ratio of the result (N) over the base (ε), both as logs to the common base (that is, 10). This is written out below (for now, curiously note but ignore the A). Thus, this shows that the DF is the ratio of the log of the number of new parts N, to the log of scale, ε with a variable A that we can usually ignore:

DF = -AlogεN
Eq. 2) DF = -A(log N / log ε)

I stuck an A in there along with N and ε, didn't I? We'll talk later.

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You Can Do It

Now that you have all that scaling and log stuff down, you can calculate some fractal dimensions.

For anything scaling like the ordinary line mentioned above, the number of new parts is always the inverse of scale:

For trivial scaling: N = ε^-1
If we notice some algebra we can use to manipulate the original equation for scaling, we get
N = ε^-DF = (ε^-1)^DF
Then we can substitute in to the right hand side of this the value of N from the trivial scaling rule to get
N = (N)^DF
Then, for trivial scaling DF = (log N)/(log N) = 1.00

For the Koch fractal shown earlier, however,

4 = (1/3)^-DF
so DF = (log 4)/(log 3) = 1.26

For the 32-segment quadric fractal you surely remember from atop this section, the pattern scales into 32 new pieces each 1/8 the size of the previous. Therefore,

32 = (1/8)^-DF so DF = log 32 / log 8 = 1.67

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Sometimes You Need a Little Help From Your Friends

The "number of pieces" referred to in the above examples is equivalent to the detail in a pattern, and, for the examples given so far, we needed only to count and measure fairly simple or at least tractable-as-long-as-you-already-know-something-about-them features to find the relationship between scale and detail.

But it is not always easy to calculate a DF this way because the relationship between scale and detail is not always readily observable. Just how much would you enjoy counting and measuring to find 32 new parts for every 1/8 scaling in a quadric fractal, for instance? Out of kindness and respect for our tolerance of tedium, therefore, our friends, the fractal analysts, have developed methods to assess the DF indirectly.

They have made ways for us to infer the value of complexity from the ratio of changing detail with changing scale (as in magnification or resolution in microscopy) approximated by some measure and assigned a number we figure is close enough to its fractal dimension and that is usually a new type of DF. In FracLac, it is the box counting dimension or Dʙ.

Approximating Using Regression

The basic equation for finding a fractal dimension from such data approximating scale and detail is nearly what we already know from the scaling rule. Since we don't have a consistent relationship between scale and number of parts, we deduce it from what we do have. We scale it several times and count (by computer, thank goodness), to find a value of N at every new scale, ε and use Equation 3:

Eq. 3) DF = -lim ε→0 [log Nε   /   log ε]

In practise, we find the Dʙ as the limit or the slope of the regression line for the log-log plot of the data. This is a very handy-dandy technique, indeed. Just what we use for N and ε and exactly how this handy technique works for us in box counting with FracLac is explained on the Box Counting page.

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