Content Introduction Construction Properties Variations Author Biography
All pictures from WinCrv

Introduction   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

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.

Properties
• 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 (using WinCrv)
• 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

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.