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

5825 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 Mandelbrot Curve   The major contribution of Benoit Mandelbrot was to open the fascinating field of the fractal geometry using facts known long before he wrote his first book about fractals: Peano Curve, Von Koch Curve, Sierpinsky Objects, Hausdorff-Besicovitch dimension, ... However, as many others, his name is also attached to an interesting fractal curve based on a simple iterator.  


Construction Back to Top

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

The fisrt iteration is obtained by dividing a straight line into three equal segments and then applying the Mandelbrot iterator.
The first iteration gives the following picture:

First Iteration in Mandelbrot Curve

The procedure is then repeated with the eight segments generated by the previous iteration.

Second Iteration in Mandelbrot Curve

In the third iteration, it's already hard to find out the way through.
Third Iteration in Cesaro Curve

Properties Back to Top

  • Curve Length

    The length of the Mandelbrot curve increases at each iteration. On each iteration, the length of the segments is divided by three and the number of segments is multiplied by eight, hence the total curve length is multiplied by 2.66666 with each iteration.

    Obviously, the length of the curve tends to infinity as the iteration number increases.

  • 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 8 segments) in the Hausdorff-Besicovitch equation gives:

      D = log(8) / log(3) = 1.8928

  • 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. The drawings change dramatically with the first iterations and then remains quite constant, whatever the iteration number.

  • Basic Geometric Figure

    Instead of starting with a straight line, the drawing can start from a square.
    Mandelbrot Curve from a Line Mandelbrot Curve from a Square

  • Style

    The original iterator can be clipped, leading to a very different drawing.
    Mandelbrot Curve with original grain Mandelbrot Curve with clipped grain

    Note that the clipped grain leads to a fractal curve with a lower fractal dimension ( D = 1.6309)

Author Biography Back to Top

Benoit Mandelbrot    Born: 20 Nov 1924 in Warsaw, Poland


Benoit Mandelbrot was largely responsible for the present interest in fractal geometry. He showed how fractals can occur in many different places in both mathematics and elsewhere in nature.

Mandelbrot was born in Poland in 1924 into a family with a very academic tradition. Mandelbrot was introduced to mathematics by his two uncles.

Mandelbrot's family emigrated to France in 1936 and his uncle Szolem Mandelbrojt, who was Professor of Mathematics at the Collège de France, took responsibility for his education. Mandelbrot attended the Lycée Rolin in Paris up to the start of World War II. He completed his studies at the Ecole Polytechnique of Paris. The, Mandelbrot went to the Institute for Advanced Study in Princeton where he was sponsored by John von Neumann.

Mandelbrot returned to France in 1955 and worked at the Centre National de la Recherche Scientific. He married Ailette Kagan and shortly after returned to the United States.

In the 1970s, with the aid of computer graphics, Mandelbrot who then worked at IBM's Watson Research Center, was able to show how Julia's work is a source of some of the most beautiful fractals known today. To do this he had to develop not only new mathematical ideas, but also he had to develop some of the first computer programs to print graphics.

His work was first put elaborated in his book Les objets fractals, forms, hasard et dimension (1975) and more fully in The fractal geometry of nature in 1982.

Mandelbrot has received numerous honours and prizes in recognition of his remarkable achievements. In 1985 Mandelbrot was awarded the 'Barnard Medal for Meritorious Service to Science'. The following year he received the Franklin Medal. In 1987 he was honoured with the Alexander von Humboldt Prize, receiving the Steinmetz Medal in 1988 and many more awards including the Nevada Medal in 1991 and the Wolf prize for physics in 1993. In 1999, Mandelbrot received the Honorary Degree of Doctor of Science from the University of St Andrews.

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

  Isanaki Sudoku 2.6b


Photo Renamer 4.3