Let $KS$ be the Koch snowflake. This fractal has an iterated function system (IFS) of the form $$ KS = \bigcup_{0 \leq k \leq 6} f_k(KS) $$ with $$ f_0(z)=\frac{1}{\sqrt{3}} e^{i\pi/2} z $$ and for $0 < k \leq 6$ $$ f_k(z)=\frac{1}{\sqrt{3}} e^{ik\pi/3} + \frac{1}{3} z. $$
This seems to be commonly known. The Webpage [1] shows this behavior. Does anybody know a reference (e.g. article in a journal) where I can found this IFS for the Koch snowflake?
I tried the following things.
- I have not found any reference by a extended web and library search.
- I talked to people working with fractals. They said, it is commonly known and should be written down somewhere, but none of them found a reference (although one did a extensive search in the library).
- I contacted the author of [1]. He said, that he has taken it from Mathworld [2].
- I looked up most of the references at the bottom of [2]. I found nothing.
- Especially, nothing can be found in Koch [3], [4] and Cesàro [5].
- Some weeks ago I posted it in a German speaking newsgroup (de.sci.mathematik). No result (reference) was found.
Edit. References, where the mentioned behavior is not found, updated.
Edit. It can also not be found in the following books:
- Barnsley, "Fractals Everywhere"
- Barnsley, "Superfractals"
- Mandelbrot, B. B., "The Fractal Geometry of Nature"
- Peitgen, Jürgens, Saupe, "Chaos and Fractals"
References:
- [1] http://www.meden.demon.co.uk/Fractals/kochsnowflake.html
- [2] http://mathworld.wolfram.com/KochSnowflake.html
- [3] Koch, H. von. "Sur une courbe continue sans tangente, obtenue par une construction géométrique élémentaire." Archiv för Matemat., Astron. och Fys. 1, 681-702, 1904.
- [4] Koch, H. von. "Une méthode géométrique élémentaire pour l'étude de certaines questions de la théorie des courbes planes." Acta Math. 30, 145-174, 1906.
- [5] Cesàro, E. "Remarques sur la courbe de von Koch." Atti della R. Accad. della Scienze fisiche e matem. Napoli 12, No. 15, 1-12, 1905. Reprinted as §228 in Opere scelte, a cura dell'Unione matematica italiana e col contributo del Consiglio nazionale delle ricerche, Vol. 2: Geometria, analisi, fisica matematica. Rome: Edizioni Cremonese, pp. 464-479, 1964.
