Acheron 2.0 Menu
Fast Track What's New in Acheron 2.0 Introduction to Fractals 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 Heighway Curve Minkowski Curve Peano Curve Square Curve Sierpinski Curve Sierpinski Objects
Feedback about Acheron 2.0 Download Counters of Acheron 2.0 Support of Acheron 2.0 Safe Use of Acheron 2.0
Visitors Counter
15891 visitors (since Jan 2010)





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.

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:
The procedure is the repeated with the squares obtained by
dividing the original square in four subsquares.
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 Back to Top
 Fractal Dimension
The fractal dimension is computed using the
HausdorffBesicovitch 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 selfsimilar
objects) in the HausdorffBesicovitch equation
gives:
D = log(4) / log(2) = 2
 SelfSimilarity
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
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




