

All pictures from WinCrv



Introduction


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


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 4sides generator to a straight line
leads to this:
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:
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.



Properties 

 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 
1  1  1  1.00 
2  1/3  4  1.33 
3  1/9  16  1.77 
4  1/27  64  2.37 
5  1/81  256  3.16 
6  1/243  1024  4.21 
...  ...  ...  ... 
10  1/19683  262144  13.31 
25  1/2.82e+11  2.81e+14  996.62 
50  1/2.39e+23  3.17e+29  1324335.72 
100  1/1.71e+47  4.02e+59  2338486807656.00 
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 preceeding iteration.
The area of the curve on the next iteration continues to increase but to
a smalller 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 
1  0  0  0.00 
2  1  1  1.00 
3  4  0.44444  1.44444 
4  16  0.19753  1.64197 
5  64  0.08779  1.72976 
...  ...  ...  ... 
10  65536  0.00152  1.79878 
15  67108864  2.64014E05  1.79997 
20  68719476736  4.57841E07  1.79999 
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
HausdorffBesicovitch 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
HausdorffBesicovitch equation gives:
D = log(4) / log(3) = 1.26185
 SelfSimilarity
This property means that every part of the curve have the same overal character
than the whole picture.



Variations
(using WinCrv)


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 foursides 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 4sides 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


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 MittagLeffler 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 MittagLeffler 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 plane
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

