1. HELGE VON KOCH Helge von Koch captured the idea of an infinite length surroundinga finite area with his socalled Koch curve. This is created http://www.bath.ac.uk/~ma1ejm/koch.html

Helge von Koch captured the idea of an infinite length surrounding a finite area with his so-called Koch curve . This is created by beginning with an equilateral triangle and adding another equilateral triangle to the middle third of each side. This bifurcation rule can be applied infinitely many times in principle, and the result is as shown: A magnification of the Koch curve will look exactly the same as the original and this property is known as self-similiarity The Koch curve can be classified as fractal because it cannot be visualised in integer dimensions: it is 'rougher' than a smooth curve or line (which has 1 dimension) so is better at 'taking up space', but is not as good at filling up space as a square (which has 2 dimensions) since it doesn't really have any area. The Koch curve is said to have a fractal dimension of around 1.2618. However, fractals are not just an abstract construction - they are also found in real-world systems such as blood vessels, the internal structure of the lungs, graphs of stock market data, and so on. Nowadays, using computer programming, mathematicians can produce very realistic images of extremely complicated structures. Patterns in nature from bacteria and fern growth, to clouds and mountains can now be created by using simple formulas which are very close to repeating themselves but never actually do. MITCHELL FEIGENBAUM BACK TO CHAOS THEORY EDWARD LORENZ BENOIT MANDELBROT

2. Poster Of Koch Helge von Koch. lived from 1870 to 1924. Koch is best known for thefractal Koch curve. Find out more at http//wwwhistory.mcs.st http://www-gap.dcs.st-and.ac.uk/~history/Posters2/Koch.html

Helge von Koch lived from 1870 to 1924 Koch is best known for the fractal Koch curve. Find out more at http://www-history.mcs.st-andrews.ac.uk/history/ Mathematicians/Koch.html

3. Mandelbrot And Koch magnified more. Helge von koch helge von Koch, captured this ideain a mathematical construction called the Koch curve. To create http://students.bath.ac.uk/ns1cb/mandelbrot.html

Mendelbrot Benoit Mandelbrot was a mathematician studying this self-similarity. One of the areas he was studying was cotton price fluctuation. He noticed the numbers that produced aberrations from the point of view of normal distribution produced symmetry from the point of view of scaling. Each particular price change was random and unpredictable. But the sequence of changes was independent on scale: curves for daily price changes and monthly price changes matched perfectly. Incredibly, analysed Mandelbrot's way, the degree of variation had remained constant over a tumultuous sixty-year period that saw two World Wars and a depression Mandelbrot analysed not only cotton prices, but many other phenomena as well. At one point, he was wondering about the length of a coastline. A map of a coastline will show many bays. However, measuring the length of a coastline off a map will miss minor bays that were too small to show on the map. Likewise, walking along the coastline misses microscopic bays in between grains of sand. No matter how much a coastline is magnified, there will be more bays visible if it is magnified more.

4. Koch-Kurve, Lindenmayer / Fraktale / Panoptikum / Peter Schenk Translate this page Top / Navigation Bild zum Thema Top / Navigation koch helge von Koch war ein schwedischerMathematiker, der 1904 der die nach ihm benannte Kurve erfunden hat. http://peter.schenk.com/panopt/fraktale/koch.htm

Koch-Kurve Allgemein Die Kochkurve mit der Dimension D= log 4 / log 3 ~ 1,26 ist ein weiteres Fraktal. Konstruiert wurde das unten ersichtliche Bild (Winkel = 60 °, 4 Iterationsschritte; Ausschnitt einer Schneeflocke) durch eine rekursive Gleichung in einem Lindenmayer -System oder eben L-System. Top Navigation Bild zum Thema Top Navigation Koch Helge von Koch war ein schwedischer Mathematiker, der 1904 der die nach ihm benannte Kurve erfunden hat. Top Navigation Lindenmayer Der Biologe Aristid Lindenmayer (1925-1989) führte aufgrund seiner Untersuchungen zum Pflanzenwachstum ein Rückkopplungsprinzip ein, das unter dem Namen L-System in die Fraktalgeometrie Einzug gehalten hat. Lindenmayer entwickelte die Theorie zu diesem L-System 1968 aufgrund seiner Erkenntnisse bezüglich der kontextfreien Chromsky-Grammatiken. Eine Definitionsgleichung besteht aus einer Folge von Zeichen der Menge: F, +, - und weiteren mit folgender Bedeutung: F = Forward respektive vorwärts, + = Drehen um einen fixen Winkel nach oben, - = Drehen um einen fixen Winkel nach unten. Diese Syntax wurde in den 80er Jahren durch Papert in der bekannten Turtle-Graphik (Programmiersprache LOGO) interpretiert.

5. Einige Der Bedeutenden Mathematiker Translate this page Kepler Johannes, 1571-1630. Klein Christian Felix, 1849-1925. koch helge von, 1870-1924.Kolmogorow Andrei Nikolajewitsch, 1903-1987. Kovalevskaya Sophia, 1850-1891. http://www.zahlenjagd.at/mathematiker.html

Einige der bedeutenden Mathematiker Abel Niels Hendrik Appolonius von Perga ~230 v.Chr. Archimedes von Syrakus 287-212 v.Chr. Babbage Charles Banach Stefan Bayes Thomas Bernoulli Daniel Bernoulli Jakob Bernoulli Johann Bernoulli Nicolaus Bessel Friedrich Wilhelm Bieberbach Ludwig Birkhoff Georg David Bolyai János Bolzano Bernhard Boole George Borel Emile Briggs Henry Brouwer L.E.J. Cantor Georg Ferdinand Carroll Lewis Cassini Giovanni Domenico Cardano Girolamo Cauchy Augustin Louis Cayley Arthur Ceulen, Ludolph van Chomsky Noel Chwarismi Muhammed Ibn Musa Al Church Alonzo Cohen Paul Joseph Conway John Horton Courant Richard D'Alembert Jean Le Rond De Morgan Augustus Dedekind Julius Wilhelm Richard Descartes René Dieudonné Jean Diophantos von Alexandria ~250 v. Chr. Dirac Paul Adrien Maurice Dirichlet Peter Gustav Lejeune Eratosthenes von Kyrene 276-194 v.Chr. Euklid von Alexandria ~300 v.Chr. Euler Leonhard Fatou Pierre Fermat Pierre de Fischer Ronald A Sir Fourier Jean-Baptiste-Joseph Fraenkel Adolf Frege Gottlob Frobenius Ferdinand Georg Galois Evariste Galton Francis Sir Gauß Carl Friedrich Germain Marie-Sophie Gödel Kurt Goldbach Christian Hadamard Jacques Hamilton William Rowan Hausdorff Felix Hermite Charles Heawood Percy Heron von Alexandrien ~60 n.Chr.

6. Helge Von Koch Lite fakta om helge von koch. helge von koch var en svensk matematiker som levde år 18701924. http://fy.chalmers.se/tp/F1projekt/1999/FractPict/koch.html

Lite fakta om Helge von Koch. Helge von Koch var en svensk matematiker som levde år 1870-1924. Vid den ringa åldern av 22 år disputerade von Koch år 1892. Drygt 10 år senare blev han professor vid Kungliga Tekniska Högskolan i Stockholm. Där arbetade han som professor år 1905 till år 1911, då han istället blev professor vid Stockholms högskola. Helge von Koch har givit namn åt den så kallade von Kochs kurva, samt von Kochs snöflinga. Även Helges syskon är väl kända. Helge är bror till Sigurd och Gerard Halfred (G.H.) von Koch. Se hur Helge von Koch såg ut.

7. Koch Niels Fabian helge von koch. helge von koch's father was Richert Vogt von koch,who had a military career, and his mother was Agathe Henriette Wrede. http://www-gap.dcs.st-and.ac.uk/~history/Mathematicians/Koch.html

Niels Fabian Helge von Koch Born: 25 Jan 1870 in Stockholm, Sweden Died: 11 March 1924 in Danderyd, Stockholm, Sweden Click the picture above to see a larger version Show birthplace location Previous (Chronologically) Next Biographies Index Previous (Alphabetically) Next Main index Helge von Koch 's father was Richert Vogt von Koch, who had a military career, and his mother was Agathe Henriette Wrede. Von Koch attended a good school in Stockholm, completing his studies there in 1887. He then entered Stockholm University. Stockholm University was the third university in Sweden and it was planned from 1865, opening in 1880 with Mittag-Leffler Von Koch spent some time at Uppsala University from 1888. He was a student of Mittag-Leffler at Stockholm University. Von Koch's first results were on infinitely many linear equations in infinitely many unknowns. In 1891 he wrote the first of two papers on applications of infinite determinants to solving systems of differential equations with analytic coefficients. The methods he used were based on those published by about six years earlier. The second of von Koch's papers was published in 1892, the year in which von Koch was awarded a doctorate for his thesis which contained the results of the two papers. Von Koch was awarded a doctorate in mathematics by Stockholm University on 26 May 1892. Garding writes in [2] that his doctoral thesis was:-

8. Koch Biography of helge von koch (18701924) Niels Fabian helge von koch. Born 25 Jan 1870 in Stockholm, Sweden http://www-groups.dcs.st-and.ac.uk/~history/Mathematicians/Koch.html

Niels Fabian Helge von Koch Born: 25 Jan 1870 in Stockholm, Sweden Died: 11 March 1924 in Danderyd, Stockholm, Sweden Click the picture above to see a larger version Show birthplace location Previous (Chronologically) Next Biographies Index Previous (Alphabetically) Next Main index Helge von Koch 's father was Richert Vogt von Koch, who had a military career, and his mother was Agathe Henriette Wrede. Von Koch attended a good school in Stockholm, completing his studies there in 1887. He then entered Stockholm University. Stockholm University was the third university in Sweden and it was planned from 1865, opening in 1880 with Mittag-Leffler Von Koch spent some time at Uppsala University from 1888. He was a student of Mittag-Leffler at Stockholm University. Von Koch's first results were on infinitely many linear equations in infinitely many unknowns. In 1891 he wrote the first of two papers on applications of infinite determinants to solving systems of differential equations with analytic coefficients. The methods he used were based on those published by about six years earlier. The second of von Koch's papers was published in 1892, the year in which von Koch was awarded a doctorate for his thesis which contained the results of the two papers. Von Koch was awarded a doctorate in mathematics by Stockholm University on 26 May 1892. Garding writes in [2] that his doctoral thesis was:-

9. [HM] Helge Von Koch's Fractalistic Snowflake Curve HM helge von koch's fractalistic snowflake curve http://mathforum.com/epigone/historia_matematica/shouzerdgleld

a topic from Historia-Matematica Discussion Group [HM] Helge von Koch's fractalistic snowflake curve post a message on this topic post a message on a new topic 10 Mar 2001 [HM] Helge von Koch's fractalistic snowflake curve , by Udai Venedem 10 Mar 2001 Re: [HM] Helge von Koch's fractalistic snowflake curve , by Julio Gonzalez Cabillon 10 Mar 2001 Re: [HM] Helge von Koch's fractalistic snowflake curve , by Michael Fried 12 Mar 2001 Re: [HM] Helge von Koch's fractalistic snowflake curve , by Daniel J. Curtin 12 Mar 2001 Re: [HM] Helge von Koch's fractalistic snowflake curve , by Udai Venedem 13 Mar 2001 Re: [HM] Helge von Koch's fractalistic snowflake curve , by Avinoam Mann 13 Mar 2001 Re: [HM] Helge von Koch's fractalistic snowflake curve , by Udai Venedem 13 Mar 2001 Re: [HM] Helge von Koch's fractalistic snowflake curve , by Daniel J. Curtin 13 Mar 2001 Re: [HM] Helge von Koch's fractalistic snowflake curve , by David Masunaga 14 Mar 2001 Re: [HM] Helge von Koch's fractalistic snowflake curve , by John Fauvel 15 Mar 2001 Re: [HM] Helge von Koch's fractalistic snowflake curve , by Dave L. Renfro

10. Eliana Argenti E Tommaso Bientinesi - Caos E Oggetti Frattali -biografia Di Niel Translate this page Niels Fabian helge von koch. Nato 25 Gennaio 1870 a Stoccolma, SveziaMorto 11 Marzo 1924 a Danderyd, Stoccolma, Svezia. Figlio http://www.webfract.it/FRATTALI/nkoch.htm

Pagina iniziale Introduzione Che cosa sono i frattali? Come si realizzano i frattali? ... Area Download CARATTERISTICHE Autosimilarità Perimetro infinito e area finita Dimensione non intera Struttura complessa a tutte le scale di riproduzione ... Dinamica caotica PERSONAGGI Niels Fabian Helge von Koch Waclaw Sierpinski Gaston Maurice Julia Benoit Mandelbrot TIPI DI FRATTALI Curva di von Kock Triangolo di Sierpinski Tappeto di Sierpinski Insieme di Mandelbrot ... Nuvole frattali FRATTALI E REALTA' ...fisiologia umana ...arte ...musica ...altri campi ... Bibliografia e indirizzi utili Niels Fabian Helge von Koch Nato: 25 Gennaio 1870 a Stoccolma, Svezia Morto: 11 Marzo 1924 a Danderyd, Stoccolma, Svezia Von Koch , pubblicato nel 1906. Si divide un segmento in tre parti uguali. Si sostituisce il segmento centrale con altri due segmenti in modo da formare un triangolo equilatero privo della base. Si ripete il procedimento indefinitamente.

11. Re: [HM] Helge Von Koch's Fractalistic Snowflake Curve By Udai Venedem Re HM helge von koch's fractalistic snowflake curve by Udai Venedem http://mathforum.com/epigone/historia_matematica/shouzerdgleld/B6D48DAE.E0F%25ve

Re: [HM] Helge von Koch's fractalistic snowflake curve by Udai Venedem reply to this message post a message on a new topic Back to messages on this topic Back to Historia-Matematica Discussion Group Subject: Re: [HM] Helge von Koch's fractalistic snowflake curve Author: venedem@wanadoo.fr Date: http://perso.wanadoo.fr/alta.mathematica/ The Math Forum

12. Caos E Oggetti Frattali -Indice - Di Eliana Argenti E Tommaso Bientinesi - Translate this page CHIUDI Vita e opere di Niels Fabian helge von koch Giuseppe PeanoWaclaw Sierpinski Gaston Maurice Julia Benoit Mandelbrot CHIUDI http://www.webfract.it/FRATTALI/indice0.htm

INDICE Pagina iniziale Contattaci Introduzione Che cosa sono i frattali? Come si realizzano i frattali? ... Frattali L-System Le caratteristiche di un frattale Autosimilarità Perimetro infinito e area finita Dimensione non intera Struttura complessa a tutte le scale di riproduzione ... Dinamica caotica Personaggi Vita ed opere di: Niels Fabian Helge von Koch Giuseppe Peano Waclaw Sierpinski Gaston Maurice Julia ... Benoit Mandelbrot Tipi di frattali Curva di von Koch Anti-fiocco di neve Quadratic Koch Triangolo di Sierpinski Sierpinski: tappeto e tappeto con tecnica l-system Frattale di Peano Frattale di Hilbert Frattale di Gosper ... Alberi frattali Alberi frattali con tecnica l-system Curva logistica Nuvole frattali Frattali e realtà I frattali in... fisiologia umana arte musica altri campi ... www.webfract.it di Eliana Argenti e Tommaso Bientinesi

13. De Kromme Van Helge Von Koch (1904) De Kromme van helge von koch (1904) Een lijnstuk wordt in drie gelijke stukken verdeeld. Het middelste stuk wordt weggelaten en er worden twee even grote lijnstukken toegevoegd. Zo ontstaat de tekening hieronder. http://home.wxs.nl/~Philip.van.Egmond/wiskunde/koch1-n.htm

De Kromme van Helge von Koch (1904) Een lijnstuk wordt in drie gelijke stukken verdeeld. Het middelste stuk wordt weggelaten en er worden twee even grote lijnstukken toegevoegd. Zo ontstaat de tekening hieronder. In het midden is een gelijkzijdige driehoek ontstaan, waarvan de basis ontbreekt. Dan krijg je de tekening, hieronder. Na nog een keer hetzelfde principe toepassen, krijg je: Ik heb dat 100 keer gedaan en dan krijg je het volgende plaatje: Veronderstel dat de lengte van het eerste lijnstuk 1 dm (=10 cm)is. Dan wordt de lengte van elk van de lijnstukken in de tweede tekening 1/3 dm. De totale lengte van die 4 lijnstukken 4*1/3= 4/3 dm. In de derde tekening zijn er al 16 lijnstukken getekend. Elke keer wordt zo'n lijnstukje 3 keer zo klein, maar je krijgt wel 4 keer zoveel lijnstukken. De totale lengte wordt dus elke keer 4/3 keer zo groot. De totale lengte van de 16 lijnstukken is dus (4/3) ~ 1,78 dm.

14. De Kromme Van Helge Von Koch (1904) De Kromme van helge von koch (1904) Een lijnstuk wordt in drie gelijkestukken verdeeld. Het middelste stuk wordt weggelaten en http://home.planet.nl/~Philip.van.Egmond/wiskunde/koch1-n.htm

De Kromme van Helge von Koch (1904) Een lijnstuk wordt in drie gelijke stukken verdeeld. Het middelste stuk wordt weggelaten en er worden twee even grote lijnstukken toegevoegd. Zo ontstaat de tekening hieronder. In het midden is een gelijkzijdige driehoek ontstaan, waarvan de basis ontbreekt. Dan krijg je de tekening, hieronder. Na nog een keer hetzelfde principe toepassen, krijg je: Ik heb dat 100 keer gedaan en dan krijg je het volgende plaatje: Veronderstel dat de lengte van het eerste lijnstuk 1 dm (=10 cm)is. Dan wordt de lengte van elk van de lijnstukken in de tweede tekening 1/3 dm. De totale lengte van die 4 lijnstukken 4*1/3= 4/3 dm. In de derde tekening zijn er al 16 lijnstukken getekend. Elke keer wordt zo'n lijnstukje 3 keer zo klein, maar je krijgt wel 4 keer zoveel lijnstukken. De totale lengte wordt dus elke keer 4/3 keer zo groot. De totale lengte van de 16 lijnstukken is dus (4/3) ~ 1,78 dm.

15. The Curve Of Helge Von Koch (1904) The curve of helge von koch (1904) A line is divided into three equal sections. Thecentral one is removed, and two lines of equal length are put in its place. http://home.planet.nl/~Philip.van.Egmond/wiskunde/koch1-e.htm

The curve of Helge von Koch (1904) A line is divided into three equal sections. The central one is removed, and two lines of equal length are put in its place. In this way the figure below is generated. In the center is now an equilateral triangle, with its lowest side removed. What would happen if you were to divide every line into three pieces, remove the central piece, and replace it with two other ones of equal length just like was done with the first line? You would see the situation below. If one did the same again: After 100 iterations I had the following picture: Now suppose that the length op the first line was 1 dm (=10cm). Then the length of the segments in the second picture would be 1/3 dm. The total length of the segments would be 4*1/3 dm=4/3dm. After the second iteration there are already 16 segments in the curve. With each iteration a segment gets 3 times as small, but 4 times as many segments are created. The total length of the 16 segments is therefore (4/3) ~ 1,78 dm.

16. Helge Von Koch Shelby Turcotte December 6, 2001 College Math Report helge von koch Niels Fabian helge von koch was born to Richert Vogt von koch and Agathe Henriette Wrede on January 25, 1870. http://www2.thomas.edu/students/t/turcottesh/Helge%20Von%20Koch.doc

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17. [HM] Helge Von Koch's Fractalistic Snowflake Curve a topic from HistoriaMatematica Discussion Group HM helge von koch's fractalisticsnowflake curve. post a message on this topic post a message on a new topic http://mathforum.org/epigone/historia_matematica/shouzerdgleld

a topic from Historia-Matematica Discussion Group [HM] Helge von Koch's fractalistic snowflake curve post a message on this topic post a message on a new topic 10 Mar 2001 [HM] Helge von Koch's fractalistic snowflake curve , by Udai Venedem 10 Mar 2001 Re: [HM] Helge von Koch's fractalistic snowflake curve , by Julio Gonzalez Cabillon 10 Mar 2001 Re: [HM] Helge von Koch's fractalistic snowflake curve , by Michael Fried 12 Mar 2001 Re: [HM] Helge von Koch's fractalistic snowflake curve , by Daniel J. Curtin 12 Mar 2001 Re: [HM] Helge von Koch's fractalistic snowflake curve , by Udai Venedem 13 Mar 2001 Re: [HM] Helge von Koch's fractalistic snowflake curve , by Avinoam Mann 13 Mar 2001 Re: [HM] Helge von Koch's fractalistic snowflake curve , by Udai Venedem 13 Mar 2001 Re: [HM] Helge von Koch's fractalistic snowflake curve , by Daniel J. Curtin 13 Mar 2001 Re: [HM] Helge von Koch's fractalistic snowflake curve , by David Masunaga 14 Mar 2001 Re: [HM] Helge von Koch's fractalistic snowflake curve , by John Fauvel 15 Mar 2001 Re: [HM] Helge von Koch's fractalistic snowflake curve , by Dave L. Renfro

18. The Curve Of Helge Von Koch (1904) The curve of helge von koch (1904) A line is divided into three equal sections. The central one is removed, and two lines of equal length are put in its place. In this way the figure below is generated. http://home.wxs.nl/~Philip.van.Egmond/wiskunde/koch1-e.htm

The curve of Helge von Koch (1904) A line is divided into three equal sections. The central one is removed, and two lines of equal length are put in its place. In this way the figure below is generated. In the center is now an equilateral triangle, with its lowest side removed. What would happen if you were to divide every line into three pieces, remove the central piece, and replace it with two other ones of equal length just like was done with the first line? You would see the situation below. If one did the same again: After 100 iterations I had the following picture: Now suppose that the length op the first line was 1 dm (=10cm). Then the length of the segments in the second picture would be 1/3 dm. The total length of the segments would be 4*1/3 dm=4/3dm. After the second iteration there are already 16 segments in the curve. With each iteration a segment gets 3 times as small, but 4 times as many segments are created. The total length of the 16 segments is therefore (4/3) ~ 1,78 dm.

19. Re: [HM] Helge Von Koch's Fractalistic Snowflake Curve By Daniel J. Curtin Re HM helge von koch's fractalistic snowflake curve by Daniel J.Curtin. reply to this message post a message on a new topic Back http://mathforum.org/epigone/historia_matematica/shouzerdgleld/p05001900b6d28dd8

Re: [HM] Helge von Koch's fractalistic snowflake curve by Daniel J. Curtin reply to this message post a message on a new topic Back to messages on this topic Back to Historia-Matematica Discussion Group Subject: Re: [HM] Helge von Koch's fractalistic snowflake curve Author: curtin@NKU.EDU Date: Mon, 12 Mar 2001 10:09:57 -0500 Dear HM, The ORESME reading group, http://www.nku.edu/~curtin/oresme.html , read these papers in 1998. There are links to translations and to scans of the four pages of additional material in the later paper on this web page. Here is an extract from our site about the texts: The paper "Sur une courbe continue sans tangente obtenue par une construction g/eome/trique e/le/mentaire,", Archiv for Matematik, Astronomi och Fysik, 1 (1904) 681-702, (trans. by Ilan Vardi in "Classics on Fractals," Gerald Edgar, ed.), in which Helge von Koch presented the Koch Snowflake, was the topic of the meeting. This material is reprinted, with some additions, in "Une me/thode ge/ome/trique e/le/mentaire pour l'e/tude de certaines questiones de la the/orie des courbes planes," Acta Mathematica, 30 (1906), 145-174. Cheers Dan Daniel J. Curtin Associate Professor Department of Mathematics and Computer Science Northern Kentucky University Highland Heights, KY 41099 USA Phone (859) 572-6348 Fax (859) 572-6097 curtin@nku.edu http://www.nku.edu/~curtin/

20. References For Koch References for the biography of helge von koch References for helge von koch. Biography in Dictionary of Scientific Biography (New York 19701990). http://www-history.mcs.st-and.ac.uk/References/Koch.html

References for Helge von Koch Biography in Dictionary of Scientific Biography (New York 1970-1990). Books: L Garding, Mathematics and Mathematicians : Mathematics in Sweden before 1950 (Providence, R.I., 1998). E Hellinger and O Toepletz, II (Leipzig, 1923-27). Main index Birthplace Maps Biographies Index History Topics ... Anniversaries for the year JOC/EFR May 2000 School of Mathematics and Statistics University of St Andrews, Scotland The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/References/Koch.html

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