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edited Aug 12 2010 at 15:55 |
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.
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edited Aug 10 2010 at 8:20 |
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] 3], [4] and Cesàro [4].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.
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.
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edited Aug 7 2010 at 10:22 |
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] and Cesàro [4].
- Some weeks ago I posted it in a German speaking newsgroup (de.sci.mathematik). No result (reference) was found.
References:
- [1] http://www.meden.demon.co.uk/Fractals/kochsnowflake.html
- [2] http://mathworld.wolfram.com/KochSnowflake.html
- [3] Koch, H. von. "Une méthode Sur une courbe continue sans tangente, obtenue par une construction géométrique élémentaire pour l'étude de certaines questions de la théorie des courbes planes.élémentaire." Acta MathArchiv för Matemat., Astron. 30och Fys. 1, 145-174681-702, 19061904.
- [4] 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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asked Aug 6 2010 at 12:18 |
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] and Cesàro [4].
- Some weeks ago I posted it in a German speaking newsgroup (de.sci.mathematik). No result (reference) was found.
References:
- [1] http://www.meden.demon.co.uk/Fractals/kochsnowflake.html
- [2] http://mathworld.wolfram.com/KochSnowflake.html
- [3] 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.
- [4] 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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