IV Dimensions

4.1 Euclid

Up to these days we have been used to think and talk in the words of traditional Euclidean geometry[01] . But many complex objects described and composed by single Euclidean sections[02] do not really reflect the characteristic of the whole real-world object, clouds and mountains respectively do not correspond to simple geometric rules.

4.1.1 Euclidean and Fractal Geometry

Fractal curves consist of infinite elements which are infinitely small and which are, because of that, not tangible. These infinite elements are the reason why the length increases to infinity at an infinitely small scale and by that makes it impossible to define a point of a fractal curve by co-ordinates or describe its position on the curve exactly after all[03].

That is the main difference to Euclidean geometry: looking at any point of an Euclidean curve, its position can be described by only one parameter, e.g. the distance from a starting point. Whereas if we define a point on a coastline by its distance from another specific starting point, we will see that the position of the resulting point strongly depends on the scale of map we are using for measurement. Measuring a specific distance, e.g. given in kilometers, from a starting point on a map with a scale of 1:50.000 will lead to a new point more distant from the starting position than measuring the same distance by walking along the coastline, which would mean using a scale of 1:1[04] . From this follows that a coastline has to be more than an Euclidean one-dimensional line, but it can not be a two-dimensional object either because it does not fill the entire plane.

4.1.2 Different Dimensions

The concept of dimension used in school mostly deals with Euclidean geometry. In short, in an E-dimensional system of co-ordinates at least E-co-ordinates are needed for defining the position of a point. Consequently a point corresponds to a zero-dimensional system of co-ordinates, points on a line to a one-dimensional, a line on a plane to a two-dimensional and finally a plane on a cube to a three dimensional system of co-ordinates - see picture 22[05].

The "topological dimension", however, proceeds from the fact that each structure can be reduced to a set of points. In this connection the disconnected set of points corresponds to the dimension zero. The dimension is then a rate of how many elements of the set of points are necessary for breaking the set: e.g. a line has the dimension of one because only one point has to be taken out for breaking the line into two pieces. This also means that fractal curves are still curves because intuitively the standard-arch is a connected set that can be separated - turned into an unconnected set - by taking out only one point. In the same sense the topological dimension of the Cantor Set is zero because there are not any two points, which are connected - to separate them no point has to be removed.

4.2 Fractal Dimension

The following pages give a summary of the characteristics and explanations of fractal dimension[06]. Visually the fractal dimension is the expression of the degree of roughness, which means how much texture an object has[07]. It also shows how fast the length of a fractal increases from one iteration to the next. Fractal dimension is not an integer in contrast to the dimension in Euclidean geometry. The complex forms of clouds, blood vessels, coastlines or mountains seem to have an unrestricted complexity, but they nevertheless have a geometric regularity, their scale-independence. That means, if we analyze the structure on different scales, we will always find the same basic elements. Fractal dimension also expresses the connection between these different scales.

There are different kinds of measurement-methods for fractal dimension, some of which will be explained on the next pages: e.g. for "true" mathematical fractals the so-called self-similarity dimension "Ds" can be measured by the increase of length from one iteration to the next. The dimension of coastlines and borders as examples of fractal curves can be measured by the structured walk-method. Finally, the box-counting method is suitable for measuring the dimension of elevations of buildings, mountains and other objects. This latter method is then used in chapter "9 Statistics" for the analysis and comparison of buildings.

• 4.2.1 Self-Similarity Dimension "Ds"
• 4.2.2 Measured Dimension "d" - Structured Walk Method
• 4.2.3 Box-Counting Dimension "Db"

4.3 Conclusion

The three different dimensions dealt with above - "Ds", "d", "Db" - may produce different values for the dimension of a curve. The reasons are:

-) Self-similar structures on all scales are only found in truly mathematical fractals.
-) Overlapping curves can only be measured by self-similar dimension, because the other two methods cannot include "overlapping" parts of the curve, which means that the resulting dimension would be lower than the self-similar dimension "Ds".
-) When the box-counting dimension method is applied to a design such as the Islamic garden layout, e.g. a grid, which is not a mathematical fractal, we will nevertheless get a non-integer value for the dimension. This is true for a certain range of box-sizes, but at a smaller scale the dimension will be reduced to 1. This indicates that beyond a certain range of scale the design shows a self-similar characteristic, but looking more closely the structure is reduced to lots of straight lines. This illustrates the range of scale beyond which self-similarity is present[08].

In the journal "Scientific American"[09] - Jürgens, Peitgen and Saupe used metaphorical comparison with languages in their article in order to explain fractal geometry and their characteristic elements. In the Western languages we know a limited number of letters of a finite alphabet, which have to be put together for a special meaning. The Chinese language consists of as many signs as concepts so that we can think of infinite elements in contrast to Western languages. Each of these characters has its own meaning. The traditional Euclidean geometry can be compared with the Western languages, where letters correspond to simple basic objects such as the line and the circle. Complex forms are then produced by bringing together these simple limited elements, and only after doing so, the object gets its meaning, whereas fractal geometry, corresponding to the Chinese language in our example, consists of infinite elements, signs or, in the case of fractals, algorithms or procedural rules, so that the meaning is found in each part of the object.

Long-lasting processes, called evolution, produce nature and its objects. This implies that there are many factors that influence an object in its development. The importance of these influences varies from scale to scale, which may lead to different dimensions. Combining different rules, defined as affine transformations, like scaling, translation or rotation, can produce such more natural-looking patterns. If these combinations moreover are produced at random, the effects are even more natural, which means that random is an important factor to get natural-looking fractal patterns.

Footnotes

[01] Euclid, Greek Eukleides, was a Greek mathematician in the 4th/3rd century B.C. He was the author of "Stoicheia", "elements", which was translated into Latin in late antiquity and is the most important mathematical education manual. He taught at the Platon Academy of Alexandria.
[02] Euclidean geometry knows only a few elements such as line, circle, planes, cubes.
[03] A fractal curve has infinte indentations which means that looking at the curve from some distance we may define a point "on" the curve but with zooming in new details, that is inlets, will come up and will show that we did not catch the curve where we expected. The point may be on the one inlet or the other which could not be distinguished on the larger scale.
[04] The length of the coastline grows by decreasing the scale of the map which makes the image more exact. Finally looking at a fractal curve, the total length of the curve cannot be given, because after infinite iterations the total length of a fractal is infinite.
[05] Voß Herbert, Chaos und Fraktale - selbst programmieren (1994), Franzis-Verlag GmbH Österreich, ISBN 3-7723-7003-9, p.315.
[06] Fractal dimension, that is to say a non-integer dimension, has been known since 1919, with a mathematical background by Hausdorff. Felix Hausdorff (1868 - 1942) was a mathematician and author. Benoit Mandelbrot, in co-operation with biologists, physicists, statisticians, technicians, astrologers and meteorologists, came to the conclusion that many phenomena which have been described insufficiently up to now are self-similar as their underlying principle. The geometrical view of them is "irregular" but often also very regular. In this connection Benoit Mandelbrot called all non-smooth sets fractals or fractal sets, but he really only meant objects that are in the one or other way self-similar. Such an object does not have any characteristic size because zooming in means that it turns into itself again over at least a certain scale and for essential characters. The fractal dimension helps to classify and describe such phenomena.
[07] Bovill Carl, Fractal Geometry in Architecture and Design (1996), Birkhäuser Bosten, ISBN 3-7643-3795-8, p.27. Fractal dimension provides a quantifiable measure of the mixture of order and surprise in a rhythmic composition.
[08] Bovill Carl, Fractal Geometry in Architecture and Design (1996), Birkhäuser Bosten, ISBN 3-7643-3795-8, p.43.
[09] Jürgens Hartmut, Peitgen Heinz-Otto, Saupe Dietmar: Fraktale - eine neue Sprache für komplexe Strukturen, Spektrum der Wissenschaft (9/1989), p.54.