All pictures are from Acheron 2.0, a free explorer of geometrical fractals. You can download Acheron 2.0 here    
Acheron 2.0 displays the following fractal curves; :
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Content
Introduction
Construction
Properties
Variations
Author Biography


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All pictures from Acheron 2.0
  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.


Von Koch Curve

Von Koch Curve
Mandelbrot Curve

Mandelbrot Curve
Minkowski Curve

Minkowski Curve
Hilbert Curve

Hilbert Curve
Cesaro Curve

Cesaro Curve
Sierpinski Curve

Sierpinski Curve
Sierpinski Objects

Sierpinski Objects
Peano Curve

Peano Curve
Squares Curve

Squares Curve
Heighway Curve

Heighway Curve


  Sample from Acheron of a Peano Curve   The Peano curves are amongst the first known fractals curves. They were described the first time in 1890 by the italian mathematician Guiseppe Peano. Some confusion seems to exist for the authorship of some fractals curves so I decided to limit on the one showed in the book from Mandelbrot. Furthermore, the drawing method used is a free interpretation of the actual one ...  


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.
Here is what you get :

First Iteration in Peano Curve
Applying the same procedure to the nine small octogon gives rise to the drawing showed here:

Second Iteration in Peano Curve
Repeated several times, this procedure leads to a very intricate but fascinating pattern. Note that repeating it more than 4 time is quite useless as the curve segments start to merge and the aesthetical aspect vanishes ...

Properties Back to Top

  • Fractal Dimension


    The fractal dimension is computed using the Hausdorff-Besicovitch equation (self-similarity method):

      D = log (N) / log ( r)

    Replacing r by three ( as each segment is divided by three on each iteration) and N by nine ( as the drawing process yields 9 smaller octogons) in the Hausdorff-Besicovitch equation gives:

      D = log(9) / log(3) = 2

  • Self-Similarity

    Looking at two successive iterations of the drawing process provides graphical evidence that this property is also shared by this curve.
    First Iteration in Peano Curve Second Iteration in Peano Curve


Variations Back to Top

All Variations described are available using Acheron 2.0

  • Iteration Level

    Four recursion levels are available.

  • Curve Style

    The Peano curve is a closed curve. The inner space can be filled to increase the contrast with the surrounding background ...
Author Biography Back to Top

Guiseppe Peano    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.

Biography From School of Mathematics and Statistics - University of StAndrews, Scotland

  Isanaki Sudoku 2.6b


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