|Acheron 2.0 displays the following fractal curves; :|
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Introduction Back to Top
The euclidean geometry uses objects that have integer topological dimensions. A line or a curve is an object that have a topological dimension of one while a surface is described as an object with two topological dimensions and a cuve as an object with three dimensions. This geometry adequately describes the regular objects but failed to be applicable when it comes to consider natural irregular shapes.
Benoit B. Mandelbrot introduced a new concepts, that he called fractals, that are useful to describe natural shapes as islands, clouds, landscapes or other fragmented structures. According to Mandelbrot, the term fractals is derived from the latin adjective fractus meaning fragmented. According to Mandelbrot, a fractal can be defined as 'a set for which the Hausdorff-Besicovitch dimension strictly exceeds the topological dimension'. This clever mathematical definition, albeit quite obscure for non-initiated people, means that a fractal curve is a mathematical function that produce an image having a topological dimension between one and two. Intuitively, fractals can be seen as curves partially filling a two-dimentional area. These curves are often described as space-filling curves.
Fractals curves exhibit a very interesting property known as self-similarity. If you observe precisely the details of a fractal curve, it appears that a portion of the curve replicates exactly the whole curve but on a different scale. Mathematicians have in fact created geometrical fractal curves long before the introduction of the fractal geometry by Benoit Mandelbrot. Some of these curves are well-known as the Von Koch's snowflake or the triangle of Sierpinsky.
Construction Back to Top
The construction of the curve is fairly simple.
This process is then repeated for the 4 segments generated at the first
iteration, leading to the following drawing in the second iteration of the
The third iteration already gives a nice picture:
Increasing the iteration number provides more detailed drawings. However,
above 8 iterations, the length of the segments becomes so small ( in fact,
close to a single pixel) that further iterations are useless, only increasing
the time of curve drawing.
The length of the Von Koch curve increases at each iteration. On each iteration, the size of the segments is divided by three and the number of segments is multiplied by four, hence a length increase by 4/3 with each iteration.
Note: the figures are valid for 'classical' Von Koch curves, for which the similarity ratio is standard.
Assuming a unit length for the starting straight line segment, we obtain the following figures:
Of course, the figures at the bottom of the table does not have any physical meaning if we speak about actually drawing such a curve as there are no physical objects of that size ... but they show a really amazing property of these curves: as the number of iteration increases, the curve length tends to infinity while it is enclosed in a finite area !!!
As said above, the Von Koch curve is enclosed in a finite area.
Putting aside the very first step of curve drawing which is a simple straigth line, we consider the area of the first equilateral triangle as a unit surface.
On the next step, four small triangles are added, one on each segment of the curve. The surface of these rectangles is one ninth of the unit surface of the triangle drawn at the preceding iteration.
The area of the curve on the next iteration continues to increase but to a smaller extent: 16 small triangles are added but their area are now 81 times smaller than the very first triangle of the Von Koch curve.
The following figures show the area increase:
Mathematically, it gives:
S = 1 + (4/9)n where n is the number of iteration after the one drawing the first triangle.
At infinite iteration, the curve approachs the limit of that equation:
S = (4/9) / ( 1 - (4/9)) = 0.8
Still, this very interesting property of the Von Koch curve: its area converges rapidly to a finite limit while the total length of the segments that composed that curve have no limit.
All Variations described are available using Acheron 2.0
The construction of the Von Koch curve allows numerous variations.
The height of the triangle generated by the drawing process is determined by the similarity ratio. The similarity ratio is expressed in 10th of percent of the line segment before the drawing process. The ratio that give a triangle height equal to the segment length is then 333.
This ratio can take value from 200 to 1000. The more below 333, the more the curve is flat and without big interest. The higher the ratio, the bigger is the curve, exploding outside its limit above 600.
The following curves were obtained using the same attributes
and applying three different similarity ratio.
Born: 25 Jan 1870 in Stockholm, Sweden
Died: 11 March 1924 in Danderyd, Stockholm, Sweden
Niels Fabian Helge von Koch attended a good school in Stockholm, completing his studies there in 1887. He then entered Stockholm University.
Von Koch spent some time at Uppsala University from 1888. He was a student of Mittag-Leffler at Stockholm University. Von Koch's first results were on infinitely many linear equations in infinitely many unknowns. In 1891 he wrote the first of two papers on applications of infinite determinants to solving systems of differential equations with analytic coefficients. The methods he used were based on those published by Poincaré about six years earlier. The second of von Koch's papers was published in 1892. Von Koch was awarded a doctorate in mathematics by Stockholm University on 26 May 1892.
Between the years 1893 and 1905 von Koch had several appointements as an assistant professor of mathematics. Von Koch was then appointed to the chair of pure mathematics at the Royal Technological Institute in Stockholm. In July 1911 von Koch succeeded Mittag-Leffler as professor of mathematics at Stockholm University.
Von Koch is famous for the Koch curve which appears in his paper Une méthode géométrique élémentaire pour l'étude de certaines questions de la théorie des courbes planes published in 1906.
The first person to give an example of an analytic construction of a function which is continuous but nowhere differentiable was Weierstrass. At the end of his paper, von Koch gives a geometric construction, based on the von Koch curve, of such a function which he also expresses analytically.
Von Koch also wrote papers on number theory, in particular he wrote several papers on the prime number theorem.