A new dimension?

In order to properly distinguish “fractal” objects from “ordinary” ones, we need to refresh a bit our ideas about the notion of “dimension”. If we ignore time, we all live in a three-dimensional space and we are pretty familiar with three-dimensional objects: a cube, a sphere, a cone, a statue, a car. From elementary school we are also familiar with certain two-dimensional objects: circles, squares, triangles and other polygons drawn on a sheet of paper are, conventionally, two-dimensional. Images on a screen (like when we watch a movie at the cinema) may represent three-dimensional objects, but they are, in fact, two-dimensional ones. A bit harder is to speak about one-dimensional or zero-dimensional objects, but still we are used to say that a line drawn on a paper or a very thin wool thread are objects of dimension one (as we are able to imagine that they have no thickness) and an infinitely small point is of dimension zero.

Informally, we may call these the “topological” (from the Greek words τόπος, ‘place, location’, and λόγος, ‘study’) dimensions of the objects.  Of course, these are mathematical abstractions: a point that really exists cannot be infinitely small and a line, to be one-dimensional, would have to be infinitely thin. Nevertheless, they are “enough” for us to do “geometry” (from the Greek word γεωμετρία, γεω- “earth” and μετράω “I measure”): they allow us to “measure the Earth”. A one-dimensional object can be measured by its length, a two-dimensional one by its area, a three-dimensional one by its volume.

However, despite being satisfactory from an intuitive point of view, the foregoing idea of dimension is insufficient to describe the nature that surrounds us. Let us consider for a moment the von Koch snowflake, which is the curve obtained by performing the process below an infinite number of times.

Despite being a mathematical abstraction, it may illustrate quite well a non-trivial problem with our intuition of dimension. As a curve, it is looks like a polygon (with an infinite number of sides!) which encases a limited area, so it may be “dominated” or “measured” with the tools of ordinary geometry by computing its perimeter and its area.

Let us start with the perimeter. If we assume that the starting triangle has all sides of length $$1$$, then it has perimeter $$3$$.
Since every smaller segment in the star at the second step has length $$\frac{1}{3}$$, the star has perimeter $$4=\frac{4}{3}\times 3>3$$.
The flake at the third step has perimeter $$\frac{16}{3}=\frac{4}{3}\times 4>4$$.
At the fourth step it has perimeter $$\frac{64}{9}=\frac{4}{3}\times \frac{16}{3}>\frac{16}{3}$$, and so on.

In general, the perimeter at the $$n$$-th step will be $$(\frac{4}{3})^n \times 3$$ and it is constantly increasing at each step of the construction. So the perimeter of the final snowflake is infinite, but still enclosing a finite area. That’s how an innocent snowflake may slip away from our ability of “measuring” nature.

Luckily, there is another way we can look at simple dimensions, which brings a mathematical significance to the value of the dimension. If we start with a line one unit long and if we magnify it by a factor of two, we get a line two units long. Magnify it by a factor of 3 and we get a line three units long. In the same way, if we magnify the unit square by two to form a square with edges twice as long, we end up with a square whose area is four times the original one. Magnify the unit square by three to have sides that are three units long, and we end up with a square made up of nine unit squares. Finally, let’s do the same with a cube of one unit size. Magnify it by a factor of two, so it is two units wide (and long, and tall) and the result is eight unit cubes. Magnify it by a factor of 3 and we get a cube made of twenty-seven unit cubes.

In all these cases, we can describe the relationship between the magnification factor $$r$$ of the “unital” object, its dimension $$D$$ and the total number $$N$$ of “unital” objects needed to fill the magnified one, with the following equation:

$N=r^D$

In view of this relationship, if we want to determine the dimension of our object we may extract it by rearranging the formula with respect to $$D$$. To do this we need a mathematical function, called the logarithm $$log$$, which enjoys a couple of very useful properties:

$$log(x\times y)=\log(x)+log(y)$$ and $$log(x^y)=y\times logx)$$

With this function at hand, we find that $$log(N) =log(r^D) =D\times log (r)$$ and hence

$D=\frac{log(N)}{log(r)}$

and this relation is true independently from the particular magnification factor we use to compute it. We may call it fractal dimension. In the particular case of the unital cube, we have seen that if we magnify by a factor of $$r=2$$, we need $$N=8=2^3$$ unital cubes to fill the magnified one. Therefore,

$D=\frac{log(N)}{log(r)}=\frac{log(2^3)}{log(2)}=\frac{3\times log(2)}{log(2)}=3$

and hence the cube is three-dimensional even with respect to this fractal dimension. Now let’s apply this idea to the von Koch snowflake. In this case, we can see that the whole snowflake is composed by gluing together three “sides” which are von Koch’s curves (as the following one).

Intuitively, in every step of the construction taking segments which are 1/3 of the original ones means taking the one fourth of the figure in the bottom left corner of the subsequent one and we need four copies of it to recover the whole figure. For instance, if we look at ___, taking a one third segment means taking _, which is the bottom left of _/\_.

At each step of the construction, we have segments which are 1/3 of the length of the previous ones and we need four times as many segments. This proportion is always preserved and, in particular, it is preserved in the final curve. Thus, if we magnify the curve by $$r=3$$, we need $$N=4$$ curves to fill it. Summing up

$D=\frac{log(N)}{log(r)}=\frac{log(4)}{log(3)}\approx 1,26$

so its fractal dimension is not an integer (the dimension of the von Koch’s snowflake is obviously the same of any one of its components).

It is important to mention that what we just described is one example of fractal dimension, which works well for theoretical objects that can be described recursively as the von Koch’s snowflake. There exist however several different fractals dimensions, which can be more or less effective depending on the context of application.

Fractals

The name ‘fractal’, from the Latin ‘fractus’ meaning broken or fragmented, was coined by Mandelbrot in his foundational essay in 1975. Since then, fractal geometry has attracted widespread, and sometimes controversial, attention.

Various attempts have been made to give a mathematical definition of a fractal, but such definitions have not proved satisfactory in a general context.

Here we avoid giving a precise definition, preferring to consider a set in Euclidean space to be a fractal if it has all or most of the following features:

1. It has a fine structure, that is: irregular details appear at arbitrarily small scales.
2. It is too irregular to be described by calculus or traditional geometrical language, either locally or globally.
3. Often it has some sort of self-similarity or self-affinity, perhaps in a statistical or approximate sense.
4. Usually its ‘fractal dimension’ (defined in some way) is strictly greater than its topological dimension.
5. In many cases of interest it has a very simple, perhaps recursive, definition.
6. Often it has a ‘natural’ appearance.

Examples of fractals abound, but certain classes have attracted particular attention. Certain self-similar fractals are especially well known:

• the von Koch snowflake,
• the Sierpinski triangle (or gasket),
• the middle-third Cantor set,
• and the Sierpinski carpet.

The last two examples are represented respectively by the image and the animation below.

Fractals that occur as attractors or repellers of dynamical systems, for example the Julia sets resulting from iteration of complex functions, have also received wide coverage. Fractal geometry is the study of sets with properties such as 1-6.

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