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Cesaro Curve in Acheron 2.0

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

Cesaro'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 Cesaro.

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