Fractals
By Els Cant, Leen Gillis, Katrien Janssens, Laetitia Parmentier

 

1. Some historical background

The groundwork for this subject was started in the early part of this century mainly by two French mathematicians, Gaston Julia(1893-1978) and Pierre Fatou. Julia, after whom some of these sets are named, was a soldier during World War I. During an offensif designed to celibrate the Kaiser's birthday, he was wounded and lost his nose. After that, he had to wear a leather mask. A great deal of work was done on this subject for several years, but later in the 1920s the study of this field almost died out. The subject was renewed in the late 1970s through the computer experiments of Dr. Benoit Mandelbrot(also French) at Yale University. In honor of Dr. Mandelbrot, one of the sets, which he explored, was named after him. Other mathematicians such as Douady, Hubbard, and Sullivan worked also on this subject exploring more of the mathematics than the applications. Since the late 1970s this subject has been at the forefront of contemporary mathematics.

Two properties of a fractal:

1. The object is self-similar and chaotic, its also based on iteration.
2. They are infinitely complex, the closer you look the more detail you see.

 

2. The H-fractal

In this fractal the 'H' is the repeated motif. The 'H'-fractal is built up from a horizontal segment which length is chosen as unit. In the first step two shorter (factor 1/sqrt(2) ) vertical segments are attached perpendicular to the end points. In the second step horizontal lines(same factor : 1/sqrt(2) ) are being attached to the 4 end points in the same way. In the third step 8 vertical lines are being added, etc. ... Here it is ended at the tenth step, where 1024 lines are added, each with a length of 1/32. For this figure the sum of all segments is : (1 + 2/(sqrt(2)) +22./2 + ...+210/25 ) = 63 + 31.(sqrt(2)) . The H-fractal is considered to be a good approximation of the ideal fractal.

The H-fractal

3. The fractal of 'Von Koch'

In 1904 the mathematician Helge von Koch gave an exemple of a curve that doesn't have a tangent anywhere.
For the mathematicians of that time a shocking experience because it meant that each part of a curve, no matter how small it is, has an infinite length.

We start with a basic segment of length 3. At first we take away the middlest part and fill it up with the upstanding parts of an equilateral triangle. In that way a broken line of 4 equally sized segments is created, the model or motif. In the next stage each of the 4 segments is seen as a base and replaced by the corresponding model on the smaller scale, etc. ... The curve of Von Koch is self-simular. Every part is a miniature of the whole.

The Koch-island originates when a curve is being placed on every side of a regular polygon. If we start with an equilateral triangle (sides with initial length of 3cm) we find that after 70 steps the total circumference of the Koch-island is 50 133 km which is larger than the circumference of our planet.  

Construction of the Von Koch-fractal and the Koch-island

 

4. The tree of Pythagoras and other fractals

 

The tree of Pythagoras

This is a construction of A.E. Bosman (1891-1961), drawn duringWorld War II. Now a computer can draw it in a few minutes.

Any square with number 'n', supports an isosceles right-angled triangle supporting itself two little squares. The left one bears the even number '2n', the right the odd number '2n+1'. These trees are an illustration of the theorem of Pythagoras : the sum of the areas of the little squares is the same as the area of the big square. So when square n has area k, then the sum of the areas of squares 2n and 2n+1 is 2 x k/2 or also k. On the picture the sum of the area of the eight squares numbered 8, 9, 10, ..., 15 is the same as the area of square 1.


Fractal made by two turn-multiplications

Tree of Pythagoras based on a stem-motive

5. The Wonder-World of Mandelbrot

The Mandelbrot set is, as mathematicians might say, the locus of points, C, for which the series Zn+1 = Zn + C starting from Z0 = (0,0) for Zn+1, Zn and C being complex numbers.

One of the fascinating things about the Mandelbrot set is the seeming contradiction in it. It is said to be the most complex object in mathematics, perhaps the most complex object ever seen. But at the same time, it is generated by an almost absurdly simple formula : multiply Z by itself and add C ; this answer becomes the new value for Z, used to perform the operation (= iteration) again and again.... So for several values of C the iteration is performed. Some C's dissapear quickly after a few iterations to infinity, others have the strange property to stay close to the center. The set of C's who seem to be attracted by the center is called the Mandelbrot-set.

In practice we see that if abs(Z) ever exceeds two, then it will very quickly head off towards infinity which means that the point is not in the Mandelbrot set These points are assigned a typical colour based on how many iterations were done before abs(Z) exceeded two. If abs(Z) doesn't exceed two after a large number of iterations, then we give up and assume that the initial point is in the Mandelbrot set. These points are typically coloured black. The black, barnacle covered pear is the Mandelbrot set - all the bands of colour outside of it seem to be simple curious artifacts that help to expose the detail of the Mandelbrot set itself.

Changing the role of Z and C, so taking C fixed and variable Z's, then we get the so-called Julia-fractals. It is intrigating to discover that C-values, who belonged to the Mandelbrot-set, lead to 'connected' Julia-sets. C's which dropped out the Mandelbrot-set produce only disconnected Julia-sets....


The Mandelbrot set

The 'baby'-Mandelbrot set
(I'ts an enlargement of the Mandelbrot set)


Julia-set for Z-1


Julia-set for 2.96 cosZ

6. Sources

Fractals, meetkundige figuren in eindeloze herhaling, Hans Lauwerier, Bloemendaal Aramith
Chaos, Fractals, and Dynamica-Computer Experiments in Mathematics, Robert L. Devaney, Addison-Wesley
Nieuwe Delta 4A, P.Gevers, J.De Langhe, H.Martens, G.Roels, H.Vercauter, R.Vermesen, Wolters-Leuven
Vuiks Verhandelingen : Fractals, Personal Computer Magazine, June 1988

http://www.archives.math.utk.edu/topics/fractals.html

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