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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"


  • [1]
  • [2]
  • [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.
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There is no reason to expect this in the papers of Koch, since his construction was for the BOUNDARY of the set you are talking about. Or more particularly for one-third of that boundary. It is a curve, made up of two parts each similar to the whole, but shrunk by factor $1/\sqrt{3}$. Or, alternatively, four parts shrunk by factor $1/3$. –  Gerald Edgar Jul 10 '13 at 16:40

5 Answers 5

Aidan Burns, "78.13 Fractal tilings", Mathematical Gazette 78 (1994), 193–196

This article describes two remarkable tilings. The first is the Koch snowflake which will only tile the plane if tiles of two (or more) different sizes are used...

Stable URL:

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Falconer's textbook on Fractal Geometry has a discussion in iterated function systems in Chapter 9.

Using the discussion in Hans Lauwrier's book Fractals: Endlessly Repeated Geometrical Figures I was able to draw "Hata's Tree-like set". I believe Von Koch snowflake is also in that book.

Hopefully you can visualize which self-similarities genrate this fractal: In these slides, the Gosper snowflake is worked out.

I personally, like the flowsnake and I've wondered how to to get it with iterated function system.

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For the boundary of the Koch snowflake, you can look at:

THE SNOWFLAKE CURVE AS AN ATTRACTOR OF AN ITERATED FUNCTION SYSTEM Demir, B. Ozdemir, Y. Saltan, M. Communications of the Korean Mathematical Society, volume 28, issue 1, 2013. pp.155-162

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In the original paper by Hutchinson "Fractals and self-similarity" in the Math. Journal of Univ. of Indiana, (1981) it is on page 727.

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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

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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 '10 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 '10 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 '10 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 '10 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 '10 at 15:58

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