Visitors Counter
9634 visitors since Jan 2010
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.



Waclaw Sierpinski, a polish mathematician, and his collegues devised several
curves that all bear his name. The most famous one, the Sierpinski Gasket, is dated back around 1916.
These curves are based on different geometrical basis but share the same construction principle.
Here they are:
 the Sierpinksi Gasket or Sierpinski Triangle
 the Sierpinski Carpet or Sierpinski Rectangle
 the Sierpinski Pentagon
 the Sierpinski Hexagon
. 

Construction Back to Top
Two drawing methods are available:
 Geometric Method
Increasing the iteration number provides more detailed drawings. However,
above 6 iterations, the size of the generated inner objects becomes so small
( in fact, close to a single pixel) that further iterations are useless, only
increasing the time of curve drawing.
 RandomBased Method
This method is also called the Chaos Method. It works the same for all the
basic figures, with the exception of the square that does not share exactly
the same behaviour ( at least in my hands ...)
To start drawing the Sierpinski triangle, select a vertex of the triangle. Then
select randomly a second vertex and draw the point that lies in the
middle of the virtual line connecting the two selected vertices. From that new
point, connect a virtual line to another vertex ( selected randomly) and draw
the midpoint of that line. Use that point as a start for the next iteration
of the drawing process.
After a while ( say, more than one thousand iterations), a ghost curve
appears, that look like the Sierpinki gasket. Then, the more points you draw,
the better the curve details appears.
The curve on the left was drawn with the geometric method. On the right, the
curve drawn with the chaos method.
Here are the figures to get the 'midpoint' for the chaos method:
Geometric Figure  Divider 

Triangle  2 
Square  3 (*) 
Pentagon  2.618033 
Hexagon  3 
Note: in my hands, the chaos method gives a slightly different Sierpinski
carpet. Perhaps, should I call this curve differently ...
Properties Back to Top
 Curve Length
Intuitively, the length of the Sierpinski gasket is the total of the length of all the segments required to draw the object.
Consider the length S of the side of the starting triangle as a unit length. The perimeter of the triangle is 3S
The first iteration creates three triangles out of one, each having a side length equal to half the side length of the triangle it came from.
The total length is then equal to :
On the next iteration, each triangle gives three smaller triangles having a side length half of that of their parent, so that the total length is then equal to
Assuming a unit length S for the side of the starting triangle, we obtain the following figures:
Iteration Number  Curve Length  Triangle Number  Curve Length 
1   1  3.00 
2   3  4.50 
3   9  6.75 
4   27  10.12 
5   81  15.19 
6   243  22.78 
7   729  34.17 
...  ...  ...  ... 
10   19683  115.33 
n  


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 null area !!!
 Curve Area
First, consider the area of the first equilateral triangle as a unit area.
During the first iteration, we get 4 inner triangles, each having an area equal to one quarter of the area of the original triangle. As the middle one
is left unpainted, the total area of the Sierpinski curve after the first iteration is 3/4 of the original area.
Applying the same process on each three remaining triangles, we get 9 painted triangles, each having one sixteenth of the original area. The total
area is now 9/16 of the original area.
The total area can be expressed as:
Area = (3/4)^{n} where n is the iteration number.
Here are the figures for the first iterations:
Iteration Number 
Triangle Number 
Triangles Area 
Curve Area 
1  1  1  1.00 
2  3  1/4  0.75 
3  9  1/16  0.5625 
4  27  1/64  0.4218 
5  81  1/256  0.3164 
...  ...  ...  ... 
10  19683  1/262144  0.0750 
15  4782969  1/268435456  0.0178 
20  1162261467  1/274877906944  0.0042 
At infinite iteration, the curve area converges towards Zero, meaning that the Sierpinski gasket have no area !!! How amazing, if you remember
that the curve length grows indefinitely as the number of iteration increases...
The other Sierpinski objects share the same properties, only the rate of the area decrease being different.
 Fractal Dimension
The fractal dimension is computed using the
HausdorffBesicovitch equation:
D = log (N) / log ( r)
 Sierpinski Triangle
Replacing N by three ( as each iteration creates three selfsimilar
triangles) and r by two ( as the sides of the triangles are divided by two)
in the HausdorffBesicovitch equation gives:
D = log(3) / log(2) = 1.5849625
 Sierpinski Rectangle
Replacing N by 8 ( as each iteration creates eight selfsimilar
rectangles) and r by three ( as the sides of the rectangles are divided by three)
in the HausdorffBesicovitch equation gives:
D = log(8) / log(3) = 1.8927892
 Sierpinski Pentagon
The ratio between an inner pentagon and its parent is:
Ratio = 2 + 2 * cos(72) = 2.618033
Replacing N by 5 ( as each iteration creates fives selfsimilar
pentagons) and r by 2.618033 ( as the ratio between pentagons from
successive iteration is 2.618033) in the
HausdorffBesicovitch equation gives:
D = log(5) / log(2.618033) = 1.6722766
 Sierpinski Hexagon
Replacing N by 6 ( as each iteration creates six selfsimilar
hexagons) and r by 3 ( as the ratio between the sides of hexagons from
successive iteration is 3) in the
HausdorffBesicovitch equation gives:
D = log(6) / log(3) = 1.6309297
 SelfSimilarity
Selfsimilarity is one of the features exhibited by fractals that is best
illustrated by the Sierpinski gasket.
Take any of the three inner triangles of a Sierpinsky gasket and magnify
it twice. The curve obtained that way is similar to the whole curve it came
from. This is what is called selfsimilarity.
On the left, the original curve. The image was then magnified twice and
the top inner triangle cut out of the drawing. It is showed on the right. The
small 'legs' at the bottom are the top vertices of the two lower inner
triangles of the original curve.
From those drawings, the selfsimilarity becomes selfevident ...
 Link with Von Koch Curve
If you take a close look to the Siepinski hexagon, you will see that the Von
Koch curve appears in the center to the drawing
If you think about the way the hexagons are drawn inside the original hexagon,
that feature will start being obvious ...
Variations Back to Top
All Variations described are available using Acheron 2.0
The construction of the Sierpinski Objects allows several variations.
 Drawing Method
Two methods are available:
 the geometric method
Based on basic geometry, it's a fast drawing method.
 the chaos method
Based on randomness, it could take a while if you like tiny details ...
See Construction for details.
 Iteration Level
This is of course a 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
As explained above ( in Construction), four basic
geometric figures can be used for the drawings: the triangle, the square,
the pentagon and the heaxagon.
Citations Back to Top
Paper Title:In situ observation of graphene sublimation and multilayer edge reconstructions
Authors: Jian Yu Huanga,1, Feng Dingb,c, Boris I. Yakobsonc,1, Ping Lud, Liang Qie, and Ju Lie,1
Reference: http://www.pnas.org/content/early/2009/06/10/0905193106.full.pdf
Paper Title:In situ observation of graphene sublimation and multilayer edge reconstructions
Authors: Jian Yu Huanga,1, Feng Dingb,c, Boris I. Yakobsonc,1, Ping Lud, Liang Qie, and Ju Lie,1
Reference: http://www.pnas.org/content/106/25/10103.full.pdf
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


