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All pictures from Acheron 2.0


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 HausdorffBesicovitch dimension strictly exceeds the topological dimension'. This clever mathematical definition, albeit quite obscure for noninitiated 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 twodimentional area. These curves are often described as spacefilling curves.
Fractals curves exhibit a very interesting property known as selfsimilarity. 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 wellknown as the Von Koch's snowflake or the triangle of Sierpinsky.



The Sierpinski curve, named from the polish mathematician Waclaw Sierpinski who originally devised it around 1912, is much less known than the other fractal objects created by Sierpinski and his coworkers as the Sierpinski gasket or the
Sierpinski Carpet. However, this curve allows beautiful variations that make it a wonderful candidate for our excursion in the world of fractals ... 

Construction Back to Top
As most of the fractal curves, the construction of the curve is based
on the recursive procedure.
The curve grain is obtained by replacing each corner of a square by a
small square placed along the diagonal axis.
The picture of the first recursion makes it easy to understand.
> This process is then repeated for the 4 corners of the figure generated at
the previous iteration. The second iteration gives this picture:
The third iteration already gives an intricate pattern that require a
much larger drawing to follow the construction rule visually. Playing with
Acheron 2.0 will help learning the construction of this curve ...
The limit curve, obtained when iteration number tends to infinity, covers
the entire area, object known as a spacefilling curve.
Properties Back to Top
 Curve Length
Take a grid with a side of length N. The total length of the curve is
the sum of the vertical and Horizontal segments ( noted S) and the
oblique segments ( noted O).
The number of Horizontal and Vertical segments is computed using the
number of segments generated at the previous recursion, the number of segments
of the recursion 1 being equal to 8 ( S^{1} = 8).
S^{Rec} = (S^{Rec1} * 4) + 8
This leads to a generalized formula for the number of Horizontal and Vertical
segments:
S^{Rec} = ((8 * 4 ^{Rec})  8) / 3
The number of oblique segments is computed using the
number of segments generated at the previous recursion, the number of segments
of the recursion 1 being equal to 12 ( O^{1} = 12).
O^{Rec} = (O^{Rec1} * 4)  4
This leads to a generalized formula for the number of Oblique segments:
O^{Rec} = ((8 * 4 ^{Rec}) + 4) / 3
The Size of the horizontal and vertical segments changes with the
recursion according to the following formula:
ls^{Rec} = N / (2^{Rec} * 4)
This is summarized in the table below:
Iteration Number 
Segment Length 
Horz & Vert Segments 
Oblique Segments 
1  N/8  8  12 
2  N/16  40  44 
3  N/32  168  172 
4  N/64  680  684 
5  N/128  2728  2732 
As expected from the principe of curve construction, the ratio of the
total number of segments between two successive iterations tends to 4.
Considering N being of unary length and by combining the different formula
showed above, we have the following figures for the total length:
Iteration Number 
Total Length of Segments 
Total Curve Length 
 Horz & Vert  Oblique 

0  1  2.1  3.12 
1  2.5  3.89  6.39 
2  5.25  7.60  12.85 
3  10.63  15.11  25.74 
4  21.31  30.18  51.50 
5  42.65  60.35  103.00 
6  85.32  120.68  206.01 
Roughly, the length is doubling at each iteration. Graphically, it gives:
 Curve Area
The number of full squares is equal to:
S^{Rec} = ((16 * 4 ^{Rec})  16) / 3
The number of half squares is equal to:
O^{Rec} = ((8 * 4 ^{Rec}) + 4) / 3
Combining these two formula gives the total number of squares:
Number^{Rec} = ((20 * 4 ^{Rec})  14) / 3)
To compute the total area, the total square number covered by the curve
is divided by the total number of squares of the grid area:
Area^{Rec} = ( N * (20 * 4 ^{Rec})  14) / (3 * 4^{Rec} * 16))
which finally simplifies to:
Area^{Rec} = N * (( 10 * 4 ^{Rec})  7) / (24 * 4 ^{Rec})
As the number of recursion increases, the terms 10 * 4 ^{Rec}
and 24 * 4 ^{Rec} become so large that the correction by 7 have
smaller and smaller influence and can be neglected. The formula tends to:
Area^{Rec} = N * ( 10 * 4 ^{Rec}) / (24 * 4 ^{Rec})
which finally simplifies to:
Area^{Rec} = N * ( 5 / 12) = N * 0.416666
The following figures illustrates this trends:
Iteration Number 
Square Number 
Square Area 
Curve Area 
1  22  N/64  0.34175 
2  102  N/256  0.39843 
3  422  N/1024  0.41211 
4  1702  N/4096  0.41553 
5  6822  N/16384  0.41638 
...  ...  ...  ... 
10  6990502  N/16777216  0.41666 
The Sierpinski curve also share the very interesting property of the
most fractals: 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 two (as on each iteration, the size of the objects is
divided by two) and N by four (as the drawing process yields 4 selfsimilar
objects) in the HausdorffBesicovitch equation
gives:
D = log(4) / log(2) = 2
The BoxCounting method gives
the same result.
Recursion  Squares Number (N)  log(N) 
Square Size (r)  log(r) 
1  22  1.3424  N/8  0.9031 
2  102  2.0086  N/16  1.2041 
3  422  2.6253  N/32  1.5051 
4  1702  3.2309  N/64  1.8062 
5  6822  3.8339  N/128  2.1072 
...  ...  ...  ... 
10  6990502  6.8445  N/6990502  3.6123 
The plot of log(N) againts log(r) gives the following picture:
Using the figures showed in the table, the linear regression yields a
slope equal to 2.0005, as expected.
A curve with a fractal dimension equal to 2 tends to cover the entire area
in which it is drawn, as effectively observed when the iteration number is
increased sufficiently.
 SelfSimilarity
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
 Iteration Level
The first iterations give richer and richer drawings. Above a given level,
the curve fills completely the drawing area, as all the curves with a fractal
dimension equal to 2.
 Curve Style
Two ways for rendering the curve are available:
 Normal
 Filled
On both style, a grid can be added (interesting for boxcounting ...)
Author Biography Back to Top
Born: 14 March 1882 in Warsaw, Poland
Died: 21 Oct 1969 in Warsaw, Poland
Waclaw Sierpinski attended school in Warsaw where his talent
for mathematics was quickly spotted by his first mathematics teacher.
This was a period of Russian occupation of Poland and despite the
difficulties, Sierpinski entered the Department of Mathematics and Physics
of the University of Warsaw in 1899. The lectures at the University were all
in Russian and the staff were entirely Russian. It is not surprising therefore
that it would be the work of a Russian mathematician, one of his teachers
Voronoy that first attracted Sierpinski.
In 1903 Sierpinski was awarded the gold medal for an essay on Voronoy's
contribution to number theory.
Sierpinski graduated in 1904 and worked for a while as a school teacher of
mathematics and physics in a girls school in Warsaw. However when the school
closed because of a strike, Sierpinski decided to go to Krakóv to study for his
doctorate. At the Jagiellonian University in Krakóv he attended lectures by
Zaremba on mathematics, studying in addition astronomy and philosophy. He
received his doctorate and was appointed to the University of Lvov in 1908.
When World War I began in 1914, Sierpinski and his family happened to be in
Russia. When World War I ended in 1918, Sierpinski returned to Lvov. However
shortly after taking up his appointment again in Lvov he was offered a post at
the University of Warsaw which he accepted. In 1919 he was promoted to
professor at Warsaw and he spent the rest of his life there.
Sierpinski was the author of the incredible number of 724 papers and 50
books. He retired in 1960 as professor at the University of Warsaw but he
continued to give a seminar on the theory of numbers at the Polish Academy
of Sciences up to 1967.
He was awarded honorary degrees from the universities Lvov
(1929), St Marks of Lima (1930), Amsterdam (1931), Tarta (1931), Sofia (1939),
Prague (1947), Wroclaw (1947), Lucknow (1949), and Lomonosov of Moscow (1967).
He was elected to the Geographic Society of Lima (1931), the Royal Scientific
Society of Ličge (1934), the Bulgarian Academy of Sciences (1936), the national
Academy of Lima (1939), the Royal Society of Sciences of Naples (1939), the
Accademia dei Lincei of Rome (1947), the German Academy of Science (1950), the
American Academy of Sciences 1959), the Paris Academy (1960), the Royal Dutch
Academy (1961), the Academy of Science of Brussels (1961), the London
Mathematical Society (1964), the Romanian Academy (1965) and the Papal Academy
of Sciences (1967).
Biography From
School of Mathematics and Statistics  University of StAndrews, Scotland


