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 Minkowski Curve   The Minkowski Curve is also called the Minkowski sausage. According to Mandelbrot, the origin of the curve is uncertain and was dated back at least to Hermann Minkowski.  


Construction Back to Top

As almost all fractals curves, the construction of the Minkowski curve is based on a recursive procedure.

At each recursion, a 8-sides generator is applied to each line segment of the curve. As the first step starst with a straigth line, it gives:

First Iteration in Minkowski Curve Note that there are 8 differents segments (and not 7, as it can be thought at first sight ..)

The same generator is applied to the 8 segments formed at the first iteration to produce a somewhat more complex curve:

Second Iteration in Minkowski Curve

The third iteration already gives a nice picture:

Third Iteration in Minkowski Curve

The first stages of the procedure modify heavily the appearance of the curve. However, quite soon, the curve remains roughly the same whatever the recursion level, only the time required to drawn the curve increases.

Propreties Back to Top

  • Curve Length

    The length of the Minkowski curve increases at each iteration. On each iteration, the length of the segments is divided by four and the number of segments is multiplied by eight, hence the total curve length is multiplied by 2 with each iteration.

    Obviously, the length of the curve tends to infinity as the iteration number increases.

  • Fractal Dimension


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

      D = log (N) / log ( r)

    Replacing r by four ( as each segment is divided by four on each iteration) and N by eight ( as the drawing process yields 8 segments) in the Hausdorff-Besicovitch equation gives:

      D = log(8) / log(4) = 1.5

  • Self-Similarity

    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

    Five recursion levels are available. Above this iteration number, the overall aspect of the curve remains essentially unaffected.

  • Basic Geometric Figure

    Instead of starting with a straight line, the drawing can start from a triangle or a square, leading to interesting curves.
    Minkowsi Curve from a Triangle Minkowski Curve from a Square
Author Biography Back to Top


Hermann Minkowski    Born: 22 June 1864 in Alexotas, Russian Empire (now Kaunas, Lithuania)
   Died: 12 Jan 1909 in Göttingen, Germany

Hermann Minkowski studied at the Universities of Berlin and Königsberg. He received his doctorate in 1885 from Königsberg. He taught at several universities, Bonn, Königsberg and Zurich. In Zurich, Einstein was a student in several of the courses he gave.

Minkowski accepted a chair in 1902 at the University of Göttingen, where he stayed for the rest of his life. At Göttingen he learnt mathematical physics from Hilbert and his associates.

By 1907 Minkowski realised that the work of Lorentz and Einstein could be best understood in a non-educlidean space. He considered space and time, which were formerly thought to be independent, to be coupled together in a four-dimensional 'space-time continuum'. This space-time continuum provided a framework for all later mathematical work in relativity.

Minkowski was mainly interested in pure mathematics and spent much of his time investigating quadratic forms and continued fractions. His most original achievement, however, was his 'geometry of numbers'.

At the young age of 44, Minkowski died suddenly from a ruptured appendix.

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

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