All pictures are from Acheron 2.0, a free explorer of geometrical fractals. You can download Acheron 2.0 here    
Acheron 2.0 displays the following fractal curves; :
Acheron 2.0 Screen Overview
Acheron 2.0 Screen Overview
    Acheron 2.0 Screen Overview
Acheron 2.0 Screen Overview
    Acheron 2.0 Screen Overview
Acheron 2.0 Screen Overview
   



Content
Introduction
Construction
Properties
Variations
Author Biography


Visitors Counter

8211 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 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 Cesaro Curve   Ernesto Cesaro, an italian mathematician, described several curves that now bears his name. The curve showed below, dated back to 1905, was used by Mandebrot in his work on fractals.
The Cesaro curve is a set of two curves with intricate patterns that fit into each other
 


Construction Back to Top

As almost all fractals curves, the construction of the Cesaro curve is based on a recursive procedure.

To draw the cruve, start with a square. Draw the first half of the four diagonals starting from the center of the square. Draw the first half of the four medians starting at the edges of the square.
The first iteration gives the following picture:

First Iteration in Cesaro Curve

The procedure is the repeated with the squares obtained by dividing the original square in four sub-squares.
The second iteration already gives an idea of the interweaving of the two curves:

Second Iteration in Cesaro Curve

The third iteration already gives a nice picture:

Third Iteration in Cesaro Curve

Intricate patterns arise on subsequent iterations. However, quite fast, the area covered by the curve increases up to the point where it occupies the whole area.

Properties Back to Top

  • Fractal Dimension


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

      D = log (N) / log ( r)

    Replacing r by two ( as the square side is divided by two on each iteration) and N by four ( as the drawing process yields four self-similar objects) in the Hausdorff-Besicovitch equation gives:

      D = log(4) / log(2) = 2

  • Self-Similarity

    Looking at two successive iterations of the drawing process provides graphical evidence that this property is also shared by this curve.
Variations Back to Top

All Variations described are available using Acheron 2.0

  • Iteration Level

    Eight recursion levels are available. Above this level of iteration, the drawing area is covered more and more completely by the curve.

Author Biography Back to Top

Ernesto Cesāro    Born: 12 March 1859 in Naples, Italy
   Died: 12 Sept 1906 in Torre Annunziata, Italy
Ernesto Cesāro studied in Naples, then in Ličge going after some time to Ecole des Mines of Ličge. He received a doctorate from the University of Rome in 1887. Cesāro held the chair of mathematics at Palermo until 1891, moving then to Rome where he held the chair until his death.

Cesāro's main contribution was to differential geometry. This is his most important contribution which he described in Lezione di geometria intrinseca (Naples, 1890). This work contains descriptions of curves which today are named after Cesāro.

In addition to differential geometry, he worked on many topics such as number theory, divergent series and mathematical physics.

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

  Isanaki Sudoku 2.6b


Photo Renamer 4.3