3.1 The “true” Mathematical Fractals

The development of this kind of fractals consists of simple rules - a starting image, the so-called initiator, is replaced by another image, the so-called generator. But nevertheless they are very complex and always strictly self-similar: it does not matter which part we analyze, it always looks exactly like a scaled down copy of the whole set. The tools to create such fractals are called iteration and feedback: Iteration means that the procedure is repeated based on the result of the previous step.

Fractals in the mathematical sense only exist at the limit point of an infinite number of generation steps. Thus a part will be an exactly scaled down copy of the whole if both, the part and the whole, contain infinite small sections. Therefore they can only be found in theory: infinite steps are only reached after infinite time. This also means that the pictures of the fractals shown on the following pages are only approximations but they will already offer the characteristics of fractals after two or three steps. That also leads us to a more abstract definition:

Fractals are objects, which are self-similar at least in a more general sense up to a certain scale - this is also true for coastlines and elevations of buildings, as I have mentioned before.

3.1.1 Cantor Set

For producing the Cantor Set[01] the initiator, a straight line of a certain length, is replaced by a generator consisting of two lines, each of the length of 1/3 of the initiator, in such a way that the new lines are located in each case at the end of the initiator. From that an open middle interval of the same length as the lines of the generator emerges but this “hole” does not include its end points - these points belong to the two outer parts, marked 1 and 2 in picture 08. This geometric rule is repeated again with the two new lines, which leads to four lines and so on.

picture 08:Cantor Set

In theory the thickness of the Cantor Set is nearly zero, but for illustration purposes I used some thickness. The hierarchy on the right gives the cascade of the number of parts at each step.
example: The planet Saturn has not only one ring but many. The rings are split several times, which was discovered by “Voyager I”. These splits are mostly very small. In addition to that the rings let the sunshine through, which means that they are transparent, like a meager mass such as the Cantor Set. The structure of the rings may be interpreted as circles with the distances of the points of the Cantor Set.[Mandelbrot Benoit B., Dr. Zähle Ulrich (editor of the German edition), Die fraktale Geometrie der Natur (1991) einmalige Sonderausgabe, Birkhäuser Verlag Berlin, ISBN 3-7643-2646-8, p.93.]

The process of constructing the Cantor Set is called coagulating, with the mass of the middle third flowing into the right and left section.

After infinite steps the result is an infinite number of clustered points with an infinitely high density, which are situated along a line. The Cantor Set is one example of curves that Benoit Mandelbrot called “monsters” because of their unearthly characteristics. The Cantor set[02]  demonstrates very clearly two important features that fractal structures have in common:

1. Self-similarity from the large to the small scale: Any part of the Cantor Set is an exactly scaled down copy of the entire Cantor Set.
2. The clustering of the points in the Cantor Set is similar to natural systems with a random factor being added - e.g. the distribution of stars displays a clustering in their layout rather than an even or strictly random distribution of features.

The distribution of rough diamonds in the earth's crust is quite similar to the distribution of stars and galaxies in the sky - Benoit Mandelbrot presented such a comparison in his book "Die fraktale Geometrie der Natur[03] . If we mark every diamond-mine and every place of discovery - old and new one - on a world-map, the density of these marks will look irregular from a great distance: some are isolated and most are concentrated in some few regions. In these regions the distribution of diamonds is not regular either. On closer observation, there are again only some isolated concentrations of diamonds within "poorer" regions, which looks similar to the whole like a scaled down copy. What we see is a two dimensional translation of the Cantor set.

This sounds quite similar to the distribution of centers or the distribution of high density of population: both offer a clustered pattern. For example a city consists of areas of high-density within regions of lower density, and on a world-map the distribution of cities again looks irregular, with large regions of no density in between.

3.1.2 Sierpinski Gasket

For producing the Sierpinski Gasket[04] , the initiator, an equilateral triangle, is replaced by a generator consisting of three equilateral triangles, each of the size of half the initiator, in such a way that the new triangles are located in each case at the three corners of the initiator. In other words an equilateral triangle is cut out in the middle. This cut-out triangle is half the size of the initiator and rotated by 180 degrees - the side-points of the triangle are defined by the midpoints of the sides of the original triangle. The same procedure is repeated for each of the three new triangles, and so on. The remaining triangles or the set of points that are left after infinite iterations is called the Sierpinski Gasket. For further details see picture 09.

picture 09:Sierpinski Gasket

The initiator is an equilateral triangle - colored black -, the generator are three equilateral triangles of half the size - colored grey.
The so-called chaos game or a walk can also produce the Sierpinski Gasket. First a basic triangle is marked by its three corners “A, B, C”, and a starting point “P0” is defined anywhere. The rule for the walk runs like that: from the starting point move in a straight line to a corner chosen at random until half the way is covered. A point called "P1" then marks this position. From there one of the three corners is chosen randomly again and the walk goes on for half the way. This position is then defined as point “P2”. After many stages, the Sierpinski Gasket can be recognized if the first points are taken away - the system has to balance itself out. That means from P0 to P4 the points are situated in the holes, only from point P5 onwards they are lying on the Sierpinski Gasket.

3.1.3 Koch Curve

The initiator of this fractal is again a line, the generator four lines of 1/3 of the initiator[05]. For their creation, the initiator-line is divided into three equal parts, with the middle part being replaced by an equilateral triangle of the side length of 1/3 of the initiator - the lower part of the triangle, however, is taken away. This procedure is then repeated for the four new lines. After infinite steps the construction leads to the Koch curve - the geometric rule for this fractal is shown in picture10 together with a description why this curve belongs to fractal geometry.

picture 10: Koch curve

The construction rule given by an instruction for a walk: for the first iteration one starts going 1/3 of the starting line then turning 60degrees to the left, moving again the distance of 1/3 and turning two times 60 degrees to the right. After having covered the distance of 1/3 one turns again twice 60 degrees to the left and finishes the walk by the distance of 1/3.

Why is the Koch curve a fractal?
1. The curve has an infinite length after infinite iterations - it is, however, a curve bound between two ends.
2. It is very rugged, so that nowhere the curve has tangents. This also means that because of this lack of tangents a differentiation is not possible.
3. Zooming in on the curve will always reveal the same structure as the whole - so it is strictly self-similar.
But these characteristics are only true for the “real” Koch curve, that is for the result after infinite steps where no straight line exists anymore: e.g. only after these infinite steps every part of the curve is strictly self-similar, representing an exact copy of the whole - otherwise at least “one” iteration would be missing. The need of theoretical infinite iterations is also the reason why it is not possible to construct the “real” Koch curve, but only the limited solution - for a computer-screen this means the number of possible pixels. Nevertheless only after a few iterations the final character can already be shown.

The Koch curve demonstrates how one can get a curve of infinite length. The original line, the initiator, may have a length equal to one. This line is replaced in the first step by four lines, which is called the number of pieces N, of 1/3 of the original length, which is called the reduction factor s. So the resulting length of the “new” curve is , which is longer than the original line =1. In the second step four other lines again replace each of the four lines. is therefore 4*4=16, of a length of 1/3 of 1/3 of the original line so this leads to the second length
, and so on:

Hence it follows that the length of the curve increases by , with n being the number of steps, that is iterations and the generator length being 1:

The Koch Island, which consists of three Koch curves placed along the sides of an equilateral triangle, is a first approach to a snowflake but also to an idealized city plan, as we will see in chapter “6 Fine Arts and City Planning”.

3.1.4 Minkowski Curve

For constructing the Minkowski curve the initiator, a line of e.g. a unity length equal to one, is replaced by a generator consisting of eight lines. These eight lines, each 1/4 of the original line, are arranged in the following manner: horizontal lines, which are kept in position, build the first fourth and the last fourth of the original line. The second fourth consists of a line turned up 90degrees, followed by a horizontal line and finally by a line moving down 90degrees again. The third fourth is constructed by a sequence of lines that is first turned down 90 degrees then moving horizontally and finally turning up 90 degrees again to connect the last fourth. This rule of construction is then repeated for all eight new lines of the first iteration, 64 lines of the second iteration, 512 lines of the third iteration and so on - see picture 11.

picture 11: Minkowski curve

If the kind of orientation of the generator is chosen by random for each line the resulting curve will be called the random Minkowski curve. The dark and the light line show the two possible orientations of the middle part of the generator.

In comparison to the Koch curve the length of this curve grows even faster from one stage of construction to the next. The length is measured by the equation , the generator again being a line with the length of one. After the first iteration the length is given by
after the second step

 and after the 10th iteration
The resulting Minkowski curve looks like a coastline that turns in and out with bays and inlets. By random the resulting curve turns out to be even more realistic - see picture 12.

picture 12: Mathematical fractals with a chance factor

Whether the middle part of the generator of the Koch curve moves up or down is chosen at random, that means = +60° or -60° and = -. Such a procedure can produce structures similar to natural coastlines:

3.1.5 Peano Curve

The initiator of the Peano curve[06]  is once more defined through a straight line and the correspondent generator consists of nine lines 1/3 of the initiator. The first line of the generator runs horizontally, the second turns up by 90degrees, then a horizontal part follows before the curve turns down again by 90 degrees. The fifth line moves back to the end of the first line without touching it. The next part of the curve heads down by 90degrees, followed by a horizontal part before it goes up again. Finally a line located in horizontal position again forms the last section - see picture 13.

picture 13: Peano curve

The length of this curve increases by , with one being the unity length of the initiator. From that the first iteration leads to a total length of , the second iteration to , the third iteration to , and so on.

The Peano curve offers a paradox. It is a curve that fills the surface it is lying on after infinite iterations. So this curve is at the same time a one-dimensional entity, a line, and somehow also a two-dimensional unit, a plane. The phenomenon we find in this structure is the fact that a one-dimensional curve, in Euclidean terminology, has a fractal dimension of two.

Some examples of space-filling curves are the distribution of arteries and blood vessels in the body, the brain, the roots of trees, the pulmonary arteries and others - they all offer a structure of tree branches that nearly fills its space.

If we interpret the curve as a street put on top of the surface of a city, every part or house of this city - the two dimensional plane - would be reached via only one way. Using the Peano curve as a street would also mean that the whole plane is taken in by the curve and that getting at the end of this street would lead along the whole, “infinite”, curve which would not be desirable. But with the aid of fractal dimension, the measure of how much the curve exceeds its base dimension, that is the measurement of the roughness, one may be able to measure how many areas of the plane are reached by the curve. This can also be useful for the planning of streets, where one wants to reach a certain number of houses via one street.


[01] Georg Cantor, a German mathematician, created the Cantor Set in 1883. He worked on the foundation of set theory.
Bovill Carl, Fractal Geometry in Architecture and Design (1996), Birkhäuser Bosten, ISBN 3-7643-3795-8, p.9.
[02] The Cantor Set once was the solution for a problem that was analyzed by computer engineers like Benoit Mandelbrot. The analogic signals of the telephonic computer-data transfer were overlapped by extraneous noises in certain intervals. So parts of the data were useless. Mandelbrot found out that these extraneous noises followed the scheme of the Cantor Set and so he could solve the problem.
[03] Mandelbrot Benoit B., Dr. Zähle Ulrich (editor of the german edition), Die fraktale Geometrie der Natur (1991) einmalige Sonderausgabe, Birkhäuser Verlag Berlin, ISBN 3-7643-2646-8, p.106.
[04] Waclaw Sierpinski (1882-1969) was a Polish mathematician - he created the Sierpinski Gasket in the year 1919.
[05] The Swedish mathematician Helge von Koch constructed the Koch curve first in 1904, again a long time before fractals were defined.
[06] The Peano curve was constructed by Giuseppe Peano in 1890.

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