|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
Start from a octogon drawn inside a square. Divide the octogon in nine
smaller octogons. Link the outer octogons two by two and then link all those
couples to the central octogon.
All Variations described are available using Acheron 2.0
Born: 27 Aug 1858 in Cuneo, Piemonte, Italy
Died: 20 April 1932 in Turin, Italy
Giuseppe Peano was born in a farmhouse about 5 km from Cuneo. He attended the village school in Spinetta and in Cuneo. His uncle soom realised that Giuseppe was a very talented child, he took him to Turin in 1870 for his secondary schooling and to prepare him for university studies.
On 29 September 1880 Peano graduated as doctor of mathematics joined the staff at the University of Turin in 1880.
The following year he discovered, and published, a method for solving systems of linear differential equations using successive approximations, independently discovered by Emile Picard. In 1888 Peano published the book Geometrical Calculus which begins with a chapter on mathematical logic. This was his first work on the topic that would play a major role in his research over the next few years and it was based on the work of SchrŲder, Boole and Peirce. This book contains the first definition of a vector space given with a remarkably modern notation and style and, although it was not appreciated by many at the time, this is surely a quite remarkable achievement by Peano.
In 1889 Peano published (in Latin !!!) his famous axioms, called Peano axioms, which defined the natural numbers in terms of sets. This was considered at once as a landmark in the history of mathematical logic and of the foundations of mathematics.
He invented 'space-filling' curves in 1890. Hausdorff wrote of Peano's result in GrundzŁge der Mengenlehre in 1914:
This is one of the most remarkable facts of set theory.
From around 1892, Peano embarked on a new and extremely ambitious project, namely the Formulario Mathematico. The project was completed in 1908 and one has to admire what Peano achieved but although the work contained a mine of information it was little used.
Even before the Formulario Mathematico project was completed Peano was putting in place the next major project of his life: finding a universal language. In fact the final edition of the Formulario Mathematico was written in this artificial language which is another reason the work was so little used.
Peano's career was therefore rather strangely divided into two periods. The period up to 1900 is one where he showed great originality and a remarkable feel for topics which would be important in the development of mathematics. His achievements were outstanding. However this feel for what was important seemed to leave him and after 1900 he worked with great enthusiasm on two projects of great difficulty which were enormous undertakings but proved quite unimportant in the development of mathematics.