The authors biography found in this site are in fact outlines of biographies from the excellent site of the School of Mathematics and Statistics - University of StAndrews, Scotland
Feel free to link to this site to get complete biographies and much more informations about the mathematicians and their works.



Von Koch Curve

Von Koch Curve
Hilbert Curve

Hilbert Curve
Minkoswki Curve

Minkowski Curve
     
Peano Curve

Peano Curve
Mandelbrot Curve

Mandelbrot Curve
Squares Curve

Squares Curve
     
Sierpinski Curve

Sierpinski Curve
Cesaro Curve

Cesaro Curve
Sierpinski Objects

Sierpinski Objects
     
  Heighway Curve

Heighway Curve
 

  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.

All the pictures come from a small Windows application WinCrv designed to draw fractal curves. The pictures show in this site and available in WinCrv were selected more for aesthetic reasons than for pure mathematical or theoretical interests. The reader is invited to read the books of Benoit Mandelbrot and other authors to get more informations about this fascinating geometry.