The Koch curve is one of the earliest fractal curves to have been described, appearing in a 1906 paper entitled "Une méthode géométrique élémentaire pour l'étude de certaines questions de la théorie des courbes plane" by the Swedish mathematician Helge von Koch (1870 - 1924) [1].
You can imagine that it was created by starting with a line segment, then recursively altering each line segment as folows:
- divide the line segment into three segments of equal length.
- draw an equilateral triangle that has the middle segment from step one as its base.
- remove the line segment that is the base of the triangle from step 2.
After doing this once you should have a shape similar to a cross section of a witch's hat.
The Koch curve is the limit which you approach as you follow the above steps over and over again.
The Koch curve has infinite length because each time you do the steps above on each line segment of your figure its length increases by one third.
The Koch snowflake is the same as the above except you start with an equilateral triangle instead of a line segment. After a few iterations it starts to look like the outline of a snowflake.
External links
- More discussion and helpful pictures (http://www.efg2.com/Lab/FractalsAndChaos/vonKochCurve.htm)
- [1] Biography of von Koch (http://www-gap.dcs.st-and.ac.uk/~history/Mathematicians/Koch.html)