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Content
Introduction
Construction
Properties
Variations
Author Biography


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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.


Von Koch Curve

Von Koch Curve
Mandelbrot Curve

Mandelbrot Curve
Minkowski Curve

Minkowski Curve
Hilbert Curve

Hilbert Curve
Cesaro Curve

Cesaro Curve
Sierpinski Curve

Sierpinski Curve
Sierpinski Objects

Sierpinski Objects
Peano Curve

Peano Curve
Squares Curve

Squares Curve
Heighway Curve

Heighway Curve


  Sample from Acheron of a Von Koch Curve   The Von Koch curves, named from the Swedish mathematician Helge Von Koch who originally devised them in 1904, are perhaps the most beautiful fractal curves. These curves are amongst the most important objects used by Benoit Mandelbrot for his pioneering work on fractals.
More than any other, the Von Koch curves allows numerous variations and have inspired many artists that produced amazing pieces of art.
 


Construction Back to Top

The construction of the curve is fairly simple.

A straight line is first divided into three equal segments. The middle segment is removed and replaced by two segments having the same length to generate an equilateral triangle. Applying such a 4-sides generator to a straight line leads to this:
First Iteration in Von Koch Curve

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 building process:
Second Iteration in Von Koch Curve

The third iteration already gives a nice picture:
Third Iteration in Von Koch Curve

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.

Properties Back to Top

  • Curve Length

    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:
    Iteration
    Number
    Segment
    Length
    Segment
    Number
    Curve Length
    1111.00
    21/341.33
    31/9161.77
    41/27642.37
    51/812563.16
    61/24310244.21
    ... ... ... ...
    101/1968326214413.31
    251/2.82e+112.81e+14996.62
    501/2.39e+233.17e+291324335.72
    1001/1.71e+474.02e+592338486807656.00

    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 !!!

  • Curve 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:
    Iteration
    Number
    Triangle
    Number
    Triangles
    Area
    Curve Area
    1000.00
    2111.00
    340.444441.44444
    4160.197531.64197
    5640.087791.72976
    ... ... ... ...
    10655360.001521.79878
    15671088642.64014E-051.79997
    20687194767364.57841E-071.79999

    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.

  • Fractal Dimension


    The fractal dimension is computed using the Hausdorff-Besicovitch equation:

      D = log (N) / log ( r)

    Replacing r by three (as each segment is divided by three on each iteration) and N by four (as the drawing process yields 4 segments) in the Hausdorff-Besicovitch equation gives:

      D = log(4) / log(3) = 1.26185

  • Self-Similarity

    This property means that every part of the curve have the same overal character than the whole picture.

Variations Back to Top

All Variations described are available using Acheron 2.0

The construction of the Von Koch curve allows numerous variations.

  • Iteration Level

    This is of course the most basic variation for the drawing of the curve. Up to 8 iterations can be performed. Above this limit, the length of the different line segments comes down close to a single pixel, meaning that any increase would not yield a significantly more detailed drawing.

  • Base Geometrical Figure

    The straight line used for the construction of the Von Koch curve can be replaced by a triangle, a square or any other geometrical figure. The four-sides generator is then applied to each side of the base figure.
    Here is what is generated with the fourth first basic geometrical figures:
    Triangle Square
    Pentagon Hexagon

  • Drawing Direction

    The 4-sides generator can be directed inwards or outwards. This can give, combined with varying ratio ( see below), very interesting patterns.
    The following curves were obtained using the same attributes
    • Recursion Level: 4
    • Similarity Ratio: 425
    • Base Figure: Square

    and then applying three different directions: inside, outside and random.
    Inside
    Outside
    Random Note: in this figure, at each step of the different iterations, the direction of drawing is choosen at random. The actual figure obtained is then always unique ...

    Note: The software always performed a curve scaling to fit the curve to the drawing area, whatever the actual area required to draw the curve. This explain the apparent difference between Inside and Outside variations.

  • Similarity Ratio

    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
    • Recursion Level: 5
    • Direction: Outside
    • Base Figure: Hexagon

    and applying three different similarity ratio.
    Value: 200 Value: 300
    Value: 400 Value: 500
    Value: 600 Value: 700
    Value: 800 Value: 900

  • Drawing Grain

    In the original Von Koch curve, the drawing grain is an equilateral triangle. The iteration process can be conducted with a square grain, leading to a quite different curve. I don't know if that curve got a name but as its drawing is similar to the original Von Koch curve, I set this grain as a variation of the original curve.
    This gives the following results:
    Iteration 1 Iteration 2
    Iteration 3 Iteration 4
Author Biography Back to Top

Niels Fabian Helge von Koch    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.

Biography From School of Mathematics and Statistics - University of StAndrews, Scotland

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