Koch snowflake

The Koch curve or Koch curve is one in 1904 imagined by the Swedish mathematician Helge von Koch example of an everywhere continuous, but nowhere differentiable curve. They also concern her one of the first fractal objects formally described. The Koch curve is one of the most cited examples of a fractal and was labeled as a monster in the discovery curve. The Koch curve is also known in the form of boiling between snowflake that is produced by a suitable combination of three Koch curves.

  • 2.1 Properties of fractal geometry
  • 2.2 Length and area
  • 2.3 Continuity and differentiability


You can see the curve graphically by means of an iterative process to construct (see Lindenmayer system). At the beginning, there is a curve from a single piece of track. The iteration consists in the fact that this route section will be replaced by another, consisting of four equally long distances stretch, which is constructed as follows: Range - 60 ° angle - Range - 120 ° angle ( in the opposite direction ) - Range - 60 ° angle - distance. Each of the four new sections has 1 /3 of the length of the original track section. In the next step each of the four sections is replaced by a stretch of the upper type.

This iteration is repeated as many times as the triangles are always to build to the same side of the curve. In this way, a sequence of distance trains, which tends towards the Koch curve.

Graphical representation of the construction

The first three iterations of the design

After five iterations:

This design principle, which iteratively, each subsection is replaced by a polygonal line, can be used also for the production of other fractal curves. For example, it is used in the dragon curve.

The design principle is closely related to the generation of the Cantor set, which is obtained if one does not replace the middle third of the track but away.

Lindenmayer system

The Koch curve can be described by a Lindenmayer system with the following properties:

  • Angle: 60 °
  • Start String:
  • Derivation rules:

Definition of the limit

The limit of this iteration ( eg, as IFS fractal ), the actual Koch curve, is in a sense infinitely fine structure and can therefore be represented only approximately graphically. In this case, the limit can easily be defined as follows:

The left endpoint of the initial distance piece is contained for example in each iteration, and thus belongs to the Koch curve. The center of the initial distance piece, however, is no longer included right from the first iteration. Another ( equally important ) limit value definition is given below by the parameter representation.


Properties of fractal geometry

The Koch curve is strictly self-similar according to their design specification, that is, it appear at any magnification over again, the same structures. It has a Hausdorff dimension of

Length and area

The length of the curve is unlimited in that the polygonal path is longer at each iteration by a factor of 4/3. After the - th iteration, the curve length is thus increased to the times.

The (top green colored ) area " below" the curve is, however, limited. When the delta is below the first iteration is the surface area of ​​1, is used in the second iteration on each of the four lines with a triangle area ninth added, and at the th iteration is added an area of. The entire area is accordingly calculated as a geometric series to

Continuity and differentiability

The curve is continuous everywhere but nowhere differentiable.

To investigate these properties, we consider the parametric representation of the - th iteration and its boundary function. If one conceives as time is that point on the polyline after the- th iteration, which is reached at the time when the polygonal path with constant speed (but with abrupt changes in direction ) passes from left to right endpoint. The functions are all continuous and converge pointwise to the limit function.

Provides you the time in a development base 4 represents, ie with the numbers 0,1,2,3, then returns the first decimal place the portion of the first construction step on, on which is located the second subsection on this in the second design step, etc. This can be the first one decimal field of magnitude construct, in which all of the following points must stop. From this property it follows that the functions even converge uniformly to. By a theorem of analysis is known as " uniform limit of continuous functions " then also continuous.

In every small portion of the curve can be found on the construction of sections that have a direction for each. Therefore, one can at any point of the curve construct a tangent, ie the curve is nowhere differentiable.

Koch snowflake

Does not begin to the process of replacing the Koch curve with a range, but with an equilateral triangle, then you get the Koch snowflake. It consists of three Koch curves and closes despite their infinite length only an area with a finite area. The Koch snowflake is not self-similar in contrast to the Koch curve.


  • An example is the fractal antenna

First publications

  • Helge von Koch: Une courbe continue sans tangent, obtenue par une construction géometrique élémentaire. Arkiv för Matematik 1 (1904 ) 681-704.
  • Helge von Koch: Une méthode pour l' étude de géométrique élémentaire certaines questions de la théorie of courbes plan. Acta Mathematica 30 (1906 ) 145-174.