# Cesaro Curve in Acheron 2.0

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Overview of Acheron 2.0
Fractals Curves in Acheron 2.0
Von Koch Curve
Mandelbrot Curve
Hilbert Curve
Cesaro Curve
Construction
Properties
Variations
Author Biography
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Minkowski Curve
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 All pictures are from Acheron 2.0, a free explorer of geometrical fractals. You can download Acheron 2.0 here Acheron 2.0 Screen Overview 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.