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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.
3.1.1 Cantor Set
For producing the Cantor Set 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.
The process of constructing the Cantor Set is called coagulating, with the mass of the middle third flowing into the right and left section.
3.1.2 Sierpinski Gasket
For producing the Sierpinski Gasket , 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.
3.1.3 Koch Curve
The initiator of this fractal is again a line, the generator four lines of 1/3 of the initiator. 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.
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
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
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 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.
 Georg Cantor, a German mathematician, created the Cantor Set in 1883. He worked on the foundation of set theory.