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. Helge Von Koch Helge von Koch. Externí spojení. 1 Strana biografie Nielse Fabiana Helgevon Koch od MacTutor historie matematiky archiv u Univerzita St Andrews. http://wikipedia.infostar.cz/h/he/helge_von_koch.html

švodn­ str¡nka Tato str¡nka v origin¡le Helge von Koch Niels Fabian Helge von Koch Leden 25 Březen 11 ) byl Å v©dÅ¡tina matematik , kdo dal jeho jm©no k slavn½ frakt¡l zn¡m½ jak Koch křivka , kter½ byl jeden nejdř­ve frakt¡l zat¡Ä­ do byli popisov¡ni. On se narodil do rodiny Å lechta Å¡v©dÅ¡tiny . Jeho dědeček, Nils Samuel von Koch (1801-1881), byl pr¡vn­k-gener¡l (" Justitiekansler ") Å v©dsko . Jeho otec, Richert Vogt von Koch (1838-1913) byl nadporuč­k-plukovn­k v Royal koňsk½ch str¡Å¾­ch Å v©dska. von Koch si zapisoval několik doklady na teorie č­sel . Jeden z jeho v½sledků byl teor©m zkuÅ¡ebn­ to Riemann hypot©za je ekvivalentn­ k zes­len© formě teor©m prvoč­sla On popsal Koch křivka v pap­r pojmenoval " Une m � thode g � om � trique � l � mentaire t©ct l ' � tude de certaines ot¡zky de la th � orie des courbes letadlo " [1]. Odkaz Plantagenet svitek krve kr¡lovsk½ (Mortimer-Percy hlasitost) mark½zem Ruvigny a Raineval (1911), strany 250 - 251 Extern­ spojen­ Strana biografie Nielse Fabiana Helge von Koch od MacTutor historie matematiky archiv u Univerzita St Andrews Toto je strojov½ překlad čl¡nku z encyklopedie Wikipedia poř­zen½ překladačem Eurotran . Cel½ text je dostupn½ za podm­nek GNU FDL licence

3. Euclide ( 3e Siècle Avant Jésus-Christ) von koch helge suédois, 1870-1924 Cemathématicien suèdois est le premier, en 1904, à exhiber à l’étonnement http://membres.lycos.fr/fractales001/histoire.html

Euclide ( 3e siècle avant Jésus-Christ) Ce mathématicien grec a rassemblé l'essentiel de la géométrie connue à son temps et a développé le principe de dimension entière de l'espace (première, deuxième, troisième). Cette géométrie, trop régulière, ne donne pas de bons modèles de la réalité. En effet, un arbre peut-il vraiment être décrit par des rectangles? Cliquer ici pour revenir où vous étiez JULIA Gaston Maurice français, 1893-1978 Né en Algérie en 1893, il fut envoyé au front français durant la première Guerre Mondiale, où il fut blessé et perdit son nez (il devra porter un masque). C'est lors de longs séjours dans les hôpitaux que ce jeune mathématicien ébauchera ses premiers travauxsur un sujet "pointu" relatifs aux fonctions complexes en prolongement de ceux commencés par Fatou : Mémoire sur l'itération des fonctions rationnelles, publié en 1918. Les ensembles de Julia ainsi découverts sont très "curieux" : ils ont un aspect fractal et seront la source des travaux de Benoît Mandelbrot des années 1970 où l'apport de l'outil informatique permit la visualisation de ces ensembles étonnants remettant en cause le concept usuel de courbe. A l'âge de 25 ans, il publie un ouvrage, "Mémoire sur l'itération des fonctions", qui fut honoré du Grand Prix de l'Académie des sciences.

4. Efg's Fractals And Chaos -- Von Koch Curve Lab Report Neils Fabian helge von koch s Snowflake . ScreenvonkochSnowflake.JPG (37028 bytes), Swedishmathematician helge von koch introduced the koch curve in 1904. http://www.efg2.com/Lab/FractalsAndChaos/vonKochCurve.htm

Fractals and Chaos von Koch Curve Lab Report Neils Fabian Helge von Koch's "Snowflake" Purpose The purpose of this project is to show how to create a von Koch curve, including a von Koch snowflake. Mathematical Background Swedish mathematician Helge von Koch introduced the "Koch curve" in 1904. Starting with a line segment, recursively replace the line segment as shown below: The single line segment in Step 0, is broken into four equal-length segments in Step 1. This same "rule" is applied an infinite number of times resulting in a figure with an infinite perimeter. Here are the next few steps: If the original line segment had length L, then after the first step each line segment has a length L/3. For the second step, each segment has a length L/3 , and so on. After the first step, the total length is 4L/3. After the second step, the total length is 4 L/3 , and after the k th step, the length is 4 k L/3 k . After each step the length of the curve grows by a factor of 4/3. When repeated an infinite number of times, the perimeter becomes infinite. For a more detailed explanation of the length computation, see [ , p. 107] or

5. Koch-Kurve, Lindenmayer / Fraktale / Panoptikum / Peter Schenk Allgemein. Die kochkurve mit der Dimension D= log 4 / log 3 ~ 1 26 ist ein weiteres Fraktal Top / Navigation. koch. helge von koch war ein schwedischer Mathematiker, der 1904 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.

6. 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:-

7. Koch Biography of helge von koch (18701924) Niels Fabian helge von koch. Born 25 Jan 1870 in Stockholm, Sweden helge von koch's father was Richert Vogt von koch, who had a military career, and 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:-

8. 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

9. Wikipedia Helge Von Koch Wikipedia Free Encyclopedia's article on 'helge von koch' Niels Fabian helge von koch (January 25, 1870 March 11, 1924) was a Swedish mathematician, who gave His grandfather, Nils Samuel von koch (1801-1881), was the Attorney-General http://rdre1.inktomi.com/click?u=http://en.wikipedia.org/wiki/Helge_von_Koch&

10. Helge Von Koch - Wikipedia, The Free Encyclopedia helge von koch. 1 A biography page of Niels Fabian helge von koch from theMacTutor History of Mathematics archive at the University of St Andrews. http://en.wikipedia.org/wiki/Helge_von_Koch

Helge von Koch Categories Mathematicians From Wikipedia, the free encyclopedia. Niels Fabian Helge von Koch January 25 March 11 ) was a Swedish mathematician , who gave his name to the famous fractal known as the Koch curve , which was one of the earliest fractal curves to have been described. He was born into a family of Swedish nobility . His grandfather, Nils Samuel von Koch (1801-1881), was the Attorney-General (" Justitiekansler ") of Sweden . His father, Richert Vogt von Koch (1838-1913) was a Lieutenant-Colonel in the Royal Horse Guards of Sweden. von Koch wrote several papers on number theory . One of his results was a theorem proving that the Riemann hypothesis is equivalent to a strengthened form of the prime number theorem He described the Koch curve in a paper entitled "Une méthode géométrique élémentaire pour l'étude de certaines questions de la théorie des courbes plane" [1]. edit Reference The Plantagenet Roll of the Blood Royal (Mortimer-Percy Volume) by the Marquis of Ruvigny and Raineval (1911), pages 250 - 251 edit External link A biography page of Niels Fabian Helge von Koch http://www-gap.dcs.st-and.ac.uk/~history/Mathematicians/Koch.html

11. [HM] Helge Von Koch's Fractalistic Snowflake Curve 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 http://mathforum.com/epigone/historia/shouzerdgleld

a topic from historia [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 The Math Forum

12. Talk:Koch Curve - Wikipedia, The Free Encyclopedia von is German for of so its helge of koch of koch alone doesn t make asmuch sense. Lirath Q. Pynnor. I thought it was common to put the von in? http://en.wikipedia.org/wiki/Talk:Koch_curve

Talk:Koch curve From Wikipedia, the free encyclopedia. Should we change references here to the " von Koch snowflake"? Dysprosia 00:12, 17 Nov 2003 (UTC) I certainly wouldn't mind; however, I think (in general) the von is dropped when not using the first name. "von is German for "of" so its Helge of Koch..."of Koch" alone doesn't make as much sense. Lirath Q. Pynnor Dysprosia 00:22, 17 Nov 2003 (UTC) There isn't a "right thing" to do; if you change it, Im not going to revert it both are used. Lirath Q. Pynnor Views Article Discussion Edit History Personal tools Log in Navigation Main Page Community portal Current events Recent changes ... Donations Search Toolbox What links here Related changes Special pages This page was last modified 00:35, 17 Nov 2003. All text is available under the terms of the GNU Free Documentation License (see for details). About Wikipedia

13. [HM] Helge Von Koch's Fractalistic Snowflake Curve 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 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

14. Hans Von Koch - ResearchIndex Document Query hans von koch scientific articles matching the query hans von koch A few years later, in 1904, helge vonkoch devised another, reminiscent of a snowflake. A few years later, in 1904, helge vonkoch devised another, reminiscent of a snowflake http://citeseer.nj.nec.com/cs?q=Hans von Koch

15. Helge Von Koch - Susning.nu helge von koch, matematiker, Niels Fabian helge von koch som var hans fullständiganamn föddes i Stockholm den 25 januari år 1870 och avled i Danderyd den http://susning.nu/Helge_von_Koch

Helge von Koch Startsida Senaste nytt Länkspegel ... Inställningar Sök: Helge von Koch matematiker , Niels Fabian Helge von Koch som var hans fullständiga namn föddes i Stockholm den 25 januari år 1870 och avled i Danderyd den 11 mars år 1924 [Bild] von Koch är mest känd för att ha skapat den fraktala kurvan von Kochs kurva samt en ännu mera berömd variant av den von Kochs snöflinga Se även: Fraktal Kaos Kaosteori [En engelsk sida om von Koch och hans kurva] Hitta mer information om samma ämne på webben På svenska ( Helge von Koch ) sök i A D G W ... SAOB A = Alltheweb , D = Dmoz , G = Google , W = Wikipedia , Y = Yahoo , NE = Nationalencyklopedin , SAOB = Svenska Akademiens ordbok Sätt betyg på den här artikeln: Startsida Senaste nytt Länkspegel Inställningar Sök: Visa andra versioner susning.nu drivs av Aronsson Datateknik Senast ändrad 13 maj 2004 (skillnad)

16. Koch Snowflake From MathWorld koch Snowflake from MathWorld A fractal, also known as the koch island, which was first described by helge von koch in 1904. It is built by starting with an equilateral triangle, removing the http://rdre1.inktomi.com/click?u=http://mathworld.wolfram.com/KochSnowflake.html

17. Von Kochs Kurva - Susning.nu von kochs kurva, presenterades av den svenske matematikern helge von koch år 1906i en publikation han kallade Une méthode géométrique élémentaire pour l http://susning.nu/Von_Kochs_kurva

Von Kochs kurva Startsida Senaste nytt Länkspegel ... Inställningar Sök: von Kochs kurva , presenterades av den svenske matematikern Helge von Koch år 1906 i en publikation han kallade " Une méthode géométrique élémentaire pour l'étude de certaines questions de la théorie des courbes plane För att skapa en kochkurva börjar man med en linje. Dela sedan linjen i tre lika långa delar. Nu ska du ha fyra lika långa delar. Upprepa proceduren för varje del för sig. (Första upprepningen utför du alltså 1 gång, nästa 1·4 gånger sedan nästa 1·4·4 ggr o.s.v.) Kochkurvan ser ut som en komplicerad kurva. Men den är oändligt lång. Om den ursprungliga linjen har längden L så har kurvan längden L × efter första upprepningen, L × efter andra, osv. Den slutliga längden går mot oändligheten. Sätter man ihop tre kochkurvor i en triangel, får man von Kochs snöflinga . Den är en sluten kurva som är oändligt lång, men har en bestämd yta! Du kan alltså inte färglägga snöflingans kant (du kan alltså inte rita den), men skulle du lyckas med det så skulle det inte vara några problem att färglägga den! [Bild på hur von Koch snöflinga utvecklas] Kochkurvans dimension är ca 1,26. Läs vidare under

18. Untitled helge von koch. kochs life. kochs Curve. koch Snowflake. helge von koch. Life of von koch http://epc.ucsc.edu/cosmos/presentations/Cl9.HelgeVonKloch.ppt

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19. Biografia Di Niels Fabian Helge Von Koch - Caos E Oggetti Frattali - Eliana Arge 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/Koch.htm

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. Precedente Successivo 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. Si ottiene una curva di tipo frattale che ha le seguenti caratteristiche : perimetro infinito, area finita, autosimilitudine, dimensione frazionaria. Si tratta inoltre di una curva continua che non ammette tangente in nessun punto. Se si parte da un triangolo equilatero e si applica questo procedimento si ottiene il " fiocco di neve " di von Koch. E' anche possibile vedere lo sviluppo del frattale quadratico di Koch con la tecnica L-system Nell' Area Download è possibile scaricare il programma che disegna il fiocco di neve scegliendo il numero di iterazioni. Indice Home Scrivi www.webfract.it di Eliana Argenti e Tommaso Bientinesi

20. Helge Von Koch Lite fakta om helge von koch. helge von koch var en svensk matematikersom levde år 18701924. Vid koch. Se hur helge von koch såg ut. 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.

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