Reference for the iterated function system of the Koch snowflake - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-22T09:37:01Z http://mathoverflow.net/feeds/question/34757 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/34757/reference-for-the-iterated-function-system-of-the-koch-snowflake Reference for the iterated function system of the Koch snowflake Daniel Krenn 2010-08-06T12:18:49Z 2010-08-12T15:55:27Z <p>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 $0 &lt; k \leq 6$ $$f_k(z)=\frac{1}{\sqrt{3}} e^{ik\pi/3} + \frac{1}{3} z.$$</p> <p>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?</p> <p>I tried the following things.</p> <ul> <li>I have not found any reference by a extended web and library search.</li> <li>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).</li> <li>I contacted the author of [1]. He said, that he has taken it from Mathworld [2].</li> <li>I looked up most of the references at the bottom of [2]. I found nothing.</li> <li>Especially, nothing can be found in Koch [3], [4] and Cesàro [5].</li> <li>Some weeks ago I posted it in a German speaking newsgroup (de.sci.mathematik). No result (reference) was found.</li> </ul> <p><strong>Edit.</strong> References, where the mentioned behavior is not found, updated. </p> <p><strong>Edit.</strong> It can also not be found in the following books:</p> <ul> <li>Barnsley, "Fractals Everywhere"</li> <li>Barnsley, "Superfractals"</li> <li>Mandelbrot, B. B., "The Fractal Geometry of Nature"</li> <li>Peitgen, Jürgens, Saupe, "Chaos and Fractals"</li> </ul> <p>References:</p> <ul> <li>[1] <a href="http://www.meden.demon.co.uk/Fractals/kochsnowflake.html" rel="nofollow">http://www.meden.demon.co.uk/Fractals/kochsnowflake.html</a></li> <li>[2] <a href="http://mathworld.wolfram.com/KochSnowflake.html" rel="nofollow">http://mathworld.wolfram.com/KochSnowflake.html</a></li> <li>[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.</li> <li>[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. </li> <li>[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.</li> </ul> http://mathoverflow.net/questions/34757/reference-for-the-iterated-function-system-of-the-koch-snowflake/34758#34758 Answer by Per Alexandersson for Reference for the iterated function system of the Koch snowflake Per Alexandersson 2010-08-06T12:30:55Z 2010-08-07T07:03:31Z <p>A quick google gives a reference in the book</p> <p><a href="http://books.google.se/books?id=HgBFW9hsGZwC&amp;lpg=PP1&amp;dq=fractals%20everywhere&amp;pg=PP1#v=onepage&amp;q=koch&amp;f=false" rel="nofollow">http://books.google.se/books?id=HgBFW9hsGZwC&amp;lpg=PP1&amp;dq=fractals%20everywhere&amp;pg=PP1#v=onepage&amp;q=koch&amp;f=false</a></p> <p>There seems to be a reference in that book, but googlebooks does not show that page...</p> <p>EDIT: The link is to the book "Chaos and fractals: new frontiers of science" by AvHeinz-Otto Peitgen,Hartmut Jürgens,Dietmar Saupe</p>