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

 Content Introduction Construction Properties Variations Author Biography

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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 Mandelbrot Curve Minkowski Curve Hilbert Curve Cesaro Curve Sierpinski Curve Sierpinski Objects Peano Curve Squares Curve Heighway 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

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:

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:

The third iteration already gives a nice picture:

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.

• 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.

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.