Per Nørgård's infinity series - and fractals

By Jørgen Mortensen

The extent to which the infinity series possesses characteristics in common with fractals is really quite striking; equally remarkable is the fact that it offered an outline of Chaos theory long before this was formulated by science.

In relation to the general description of fractals below, it may be said that the infinity series and fractals share the following qualities:

Fractals in general

The term 'fractal' is connected with the ideas of 'fracture surface' and 'break'. Whilst geometry universally works with straight lines or smooth curves, nature would seem to avoid these regularities. Take the uneven bark of a tree, jagged mountain tops, or an irregular coastline. In fact, completely regular or straight lines do not seem to appear in nature at all. The only exception is a beam of light.

A coastline is always uneven. It represents both 'self-similarity' and 'scale invariance', because from a distance one sees the same uneven picture as close to.
Scale
 In fact, one cannot properly define the length of a stretch of coastline, as this depends on the nature of the scale used. If we use a scale - in the literal sense, a measuring stick - 1 kilometre long, we will obtain a particular result, but the figure will be much higher if we use a scale of only one metre. Even though the coastline remains the same, using the smaller scale we are able to get into all the small bays and nooks and crannies.

This fact - that one cannot get 'a definitive answer' - but can continue to descend into finer and finer details, is one of the characteristics of fractal geometry.

Fractal geometry is actually a term from the beginning of the last century, but it was not until computers achieved a suitable capacity around 1980 that people really became aware of the existence of fractals and thus of a completely new scientific description of the world: Chaos theory (see, for example, James Gleick: CHAOS: Making a New Science, Penguin, USA, 1988).

In 1979, the mathematician, Benoit Mandelbrot, made computer drawings of a series of points that turned out to be one connected area, the famous Mandelbrot set.
The Mandelbrot set
 This fascinating picture is produced by investigating every point on an xy plane corresponding to a pair of numbers, for example, x = -0.12 and y = 0.74. This pair of numbers is tested to see what will happen when it is passed through a formula. The new pair of numbers that emerges from the formula is put through the formula again, and the same with the next new pair of numbers, and so on ...

    (x,y) --> FORMULA --> (xny, yny)

This is the process of iteration.

If it is the case that the orbit of the pair at some point goes to infinity, then it will not belong to the set. If - no matter how often we send it through the formula - it remains bounded, it belongs to the set. In fact, the pair -0.12 0.74 remains bounded.


In other words, the computer graphics are produced by iteration.
Iteration
The formula used to create this picture is very simple, and for this reason the wealth of detail is surprising.

Drawing boundaries on the picture is just as difficult as in the case of a coastline:
One can continue to enlarge the picture (change the scale) and see more details in the border areas. The exciting thing about these details is that they reveal a similarity with the whole picture. Again and again one discovers small Mandelbrot figures; the Mandelbrot set reveals self-similarity. It also reveals scale invariance, in that we see the same figure in different scales or sizes (see, for example, Peder Voetman Christiansen Den fraktale uendelighed (Fractal Infinity) in Paradigma No. 4, 1987). New figures are also constantly appearing, including slightly altered Mandelbrot sets, revealing, that is, a partial likeness.
Drawing boundaries
Self-similarity
Scale invariance
Unpredictable?
The pictures reveal an overwhelming richness of detail, details that cannot be studied until the computer has finished its work. There are no short cuts: incredible as it may seem, the pictures are unpredictable, even though every detail of them can be calculated.
Recursion
Another type of fractal picture can be formed on the basis of another principle, that of recursion. Recursion typically involves carrying out a 'replication procedure' again and again, for example, of a line, which quickly develops into a 'kite curve' (see, for example, Hans Lauwerier Fractals, Images of Chaos, Princeton 1991).
Recursion can also consist in adding extra lines to a figure in finer and finer detail. This is the case with 'The tripartite tree' (p 12 in Lauwerier). Here we have three straight lines that meet, forming a basic shape that is replicated again and again. First it is expanded using 6 somewhat shorter lines, so that we get three repetitions of the basic shape on a smaller scale, using 9 straight lines in all. The next replications produce 27, 81, 243, and 729 straight lines.
The tripartite tree