David Makin's Fractal Blog

The Cantor Set

Background Maths:
Probably the most important principle required for an understanding of the Cantor Set and many other fractals is the notion of "limits", that is that a given mathematical construction (though infinite in nature) may have a limited final value.

For example, let's say you have 1 unit of distance to walk to reach a tree and you decide to reach it by covering half the remaining distance at each step, so:

The first step would take you 1/2 unit, leaving 1/2 unit. 1/2=(1/2)^1
The second step would take you 1/4 unit, leaving 1/4 unit. 1/4=(1/2)^2
The third step would take you 1/8 unit, leaving 1/8 unit. 1/8=(1/2)^3

The nth step would take you (1/2)^n unit, leaving (1/2)^n unit.

In your n steps you have covered 1/2+1/4+1/8+....+(1/2)^n units though you haven't reached the tree.

In fact if you take an infinite number of steps following the same rule of covering half the remaining distance at each step then on the infinite step you've travelled 1-(1/2)^infinity units leaving (1/2)^infinity units to go.

But of course (1/2)^infinity is zero therefore on the infinite step you reach your goal of having travelled one unit in total hence the sum of (1/2)^n for all n from 1 to infinity is one! In fact there is a similar limit for every fractional value f between 0 and 1, ie. for any fraction f between 0 and 1, the sum of f^n for all n from 1 to infinity has a limit value.

This notion of limits can also be applied to geometrical progressions as well as numerical ones and this is the case for the Cantor Set.

Construction - The construction of the Cantor Set is very simple, take a line segment:

Cantor set example 1

Remove the middle third:

Cantor set example 1

Now repeat the process, ie. remove the middle third of the two line segments, leaving 4 line segments:

Cantor set example 1

Do the same again:

Cantor set example 1

And again:

Cantor set example 1

As you should see from the above, if this process is repeated ad nauseam then the result will be a disjoint set of points spaced in a defined pattern, this set of points is the limit set of the above process and is called the Cantor Set.

One of the properties of the Cantor Set is that it is 100% self-similar at different scales eg. if you magnify the first or last third of the Cantor Set by a factor of 3 the result is identical to the whole, if you magnify the first, third, seventh or ninth ninths of the Set by a factor of 9 then again the result is identical to the whole etc.

This type of self-similarity at different scales is part of what defines a fractal, though for many fractals the self-similarity is not 100% (unlike the Cantor Set).

For some fractals (such as many linear IFS) the self similarity is only self-affine (ie. there is self-similarity at different scales but it includes stretching and/or rotation etc.), for some fractals the self-similarity at different scales is general rather than in detail (such as Mandelbrot/Julia fractals) and for other fractals the self similarity at different scales is restricted to merely statistical similarity (such as fBm/plasma fractals).

With classical mathematical objects such as a point or a line or an area or a volume we are used to the "dimension" of the object ie. a point has zero dimensions, a line one dimension, an area two, a volume three etc. but what is the dimension of the Cantor Set? Clearly it is more than a point but less than a line so its dimension should be somewhere between the two, this means of course that it must have a fractional dimension somewhere between zero and one.

Without going into great detail on the maths involved in "Measure Theory" it tells us that in fact the dimension of the Cantor Set is log(2)/log(3) or around 0.63 because at each stage of the construction if a new line segment is taken as being of unit length, then the parent line segment was 3 units long and it is replaced by line segments 2 units long in total and Measure Theory says that the dimension is log(new length)/log(old length)=log(2)/log(3).

When creating the Cantor Set consider for instance going directly to the second stage from the original line segment (ie. producing 4 new line segments) in this case if each new line segment is taken as being one unit in length then the original was 9 units long and we've now got a total of 4 units remaining, giving a dimension of log(4)/log(9) which is in fact the same dimension since log(4)/log(9)=2*log(2)/(2*log(3))=log(2)/log(3).

In actual fact there are different ways of calculating/defining the dimension of fractals and they have different names but here we will simply refer to the dimension of fractals as the "fractal dimension" ie. the fractal dimension of the Cantor Set is log(2)/log(3).

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