MathOverflow will be down for maintenance for approximately 3 hours, starting Monday evening (06/24/2013) at approximately 9:00 PM Eastern time (UTC-4).

## Reference for the iterated function system of the Koch snowflake

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. 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.
-

A quick google gives a reference in the book

There seems to be a reference in that book, but googlebooks does not show that page...

EDIT: The link is to the book "Chaos and fractals: new frontiers of science" by AvHeinz-Otto Peitgen,Hartmut Jürgens,Dietmar Saupe

-
 Checking my copy of Barnsley's book (copyright 1988, I am unsure if there has been a newer edition), there is no mention of the snowflake or Helge von Koch anywhere in the book. :( – J. M. Aug 6 2010 at 14:02 Yeah, but, the link, despite the url, links to the book "Chaos and fractals: new frontiers of science" by AvHeinz-Otto Peitgen,Hartmut Jürgens,Dietmar Saupe I started searching in Fractals Everywhere, but ended up at the book above. – Per Alexandersson Aug 7 2010 at 7:03 Hm, maybe I missed it, but I have not found something concerning the IFS of the Koch snowflake in the book "Chaos and Fractals". Where exactly did you see that? (or where is the hint for the not seen reference?) – Daniel Krenn Aug 7 2010 at 9:42 I know look carefully through a hardcopy of the book "Chaos and Fractals" and I have not found the IFS of the Koch snowflake or anything related pointing in this direction. – Daniel Krenn Aug 11 2010 at 18:05 I have both the first and second editions of "Chaos and Fractals"; there is a mention of the snowflake as an L-system, but nothing as an IFS. – J. M. Aug 12 2010 at 15:58