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Acheron 2.0 displays the following fractal curves; : | |||||||
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Introduction Back to Top
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.
Construction Back to Top As almost all fractals curves, the construction of the Cesaro curve is
based on a recursive procedure.
All Variations described are available using Acheron 2.0
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. Biography From School of Mathematics and Statistics - University of StAndrews, Scotland |