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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  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 Heighway Curve   The american physicist John Heighway discovered this curve while working at NASA. The Heighway curve, also called the Heighway Dragon or Harter-Heighway Dragon, was popularized in 1967 when Martin Gardner described it in American Scientific. Three years later, the first mathematical analysis was published by Chandler Davis and Donald Knuth. Although the construction of this curve is quite simple, its fascinating properties have stimulate a lot of works in different fields of mathematics.  


Construction Back to Top

While three methods were described for the construction of the Dragon curve, the discussion will be limited to the method used originally by Heighway.

To construct the curve, imagine that you take a trip of paper. Fold it in two equal parts by taking the right edge and adjusting it above the left edge of the trip. Repeat this operation two or three times and then unravel the paper trip so that each corner makes a right angle in the natural direction of the fold. Looking at the paper trip from the edge will reveal the Heighway curve.

Here are the result of the first five foldings:


Note that the software strips slightly the folding corners to avoid touches between folders.

Increasing the number of folding leads to more and more complex drawing and, finally, to the limiting shape, the Dragon Curve.

There are two types of folding: the folding of type 1 consists of ajusting the right edge above the left edge of the paper. The folding of type 0 is the inverse operation, as you adjust the right edge below the left one.

Mixing folding types changes dramatically the look of the curve:

Folding 11111111 ( 8 foldings of type 1)
Folding Sequence: 11111111

Folding 10101010 ( 8 foldings alternatively of type 1 and 0)
Folding Sequence: 10101010

Folding 11100111 ( 6 foldings type 1 with 2 of type 0 in the middle)
Folding Sequence: 11100111
Playing around with the folding sequence may lead to very peculiar drawings ...

Properties Back to Top

  • Curve Order


    This property derives from the construction method: a curve of order n is the combination of two curves of order n - 1.

    As an example, the following picture shows:
    • on the left, a curve obtained by using the folding sequence 111111
    • on the right, two instances of a curve obtained with a 11111 sequence, one of them being rotated to be in the correct alignment.




  • Fractal Dimension


    The fractal dimension is computed using the Hausdorff-Besicovitch equation:

      D = log (N) / log ( r)

    The self-similarity N and r values are derived from the construction of the curve using the triangle method.

      D = log(2) / log(2*sqr(2)) = 2

Variations Back to Top

All Variations described are available using Acheron 2.0

  • Folding Sequence

    The folding sequence is entered as a string of 1 and 0, up to 16 characters.

Author Biography Back to Top

  Isanaki Sudoku 2.6b


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