61. Koch Translate this page Niels Fabian helge von koch. Nasceu 25 Jan 1870 em Stockholm, Suécia.Morreu 11 Mar 1924 em Stockholm, Suécia. helge koch ( ver http://jurere.mtm.ufsc.br/~dorini/koch.html

Niels Fabian Helge von Koch Nasceu: 25 Jan 1870 em Stockholm, Suécia. Morreu: 11 Mar 1924 em Stockholm, Suécia. Helge Koch ( ver foto ) foi estudante de Mittag Leffer (também matemático - ver ref2) e o sucedeu em 1911 na Stockholm University / Suécia. Ele é famoso pela curva entitulada com seu nome ( ver curva A curva de Koch possui importantes propriedades; por exemplo: O comprimento de arco entre quaisquer dois pontos da curva é infinito. Vale salientar que a curva é limitada no plano; A área delimitada pela curva é 8/5 da área do triângulo original (a demonstração desse fato não deve ser dificil - é necessário somar a área de todos os triângulos que são acrescentados - certamente teremos uma soma infinita); A curva não admite tangente em qualquer de seus pontos; É uma curva fractal de dimensão fractal log4/log3, que é aproximadamente 1.2618. Fato curioso esse. Referências: www-groups.dcs.st-and.ac.uk/~history/Mathematicians/Koch.html www-groups.dcs.st-and.ac.uk/~history/Mathematicians/Mittag-Leffler.html

62. What (Koch's Snowflake) The koch Curve was studied by helge von koch in 1904. When considered in its snowflakeform (see below) the curve is infinitely long but surrounds finite area. http://www.ecu.edu/si/cd/interactivate/activities/koch/what.html

What is the Koch's Snowflake Activity? This activity allows the user to step through the generation of a fractal made from deforming a line by bending it. This activity allows the user to step through the generation of a fractal made from deforming a line by bending it. The Koch Curve was studied by Helge von Koch in 1904. When considered in its snowflake form (see below) the curve is infinitely long but surrounds finite area. To build the original Koch curve, start with a line segment 1 unit long. (Iteration 0, or the initiator) Replace each line segment with the following generator: Note that we are really taking the original line segment and replacing it with four new segments, each 1/3 the length of the original. Repeat this process on all line segments. Stages 0, 1 and 2 are shown below. The limit curve of this process is the Koch curve. It has infinite length. Notice also that another feature that results from the iterative process it that of self-similarity, i.e., if we magnify or "zoom in on" part of the Koch curve, we see copies of itself. The idea of the Koch curve was extended to the Koch "Snowflake" by applying the same generator to all three sides of an equilateral triangle; below are the first 4 iterations.

63. Koch Snowflakes magnified. A simple fractal is the koch snowflake, named after Swedishmathematician helge von koch (1870—1924). The construction http://www.simpson.edu/~math/labs/snow/snow.html

Koch Snowflakes n th stage snowflake? n th state snowflake? n th stage snowflake? n th stage snowflake where the final term of the sum is a function of n c) Use the command "sum(f(n),n=1..infinity);" where f n ) is the n th term of the sum for the area. This will give the area of the infinity stage Koch triangle.

64. Les Fractales Translate this page En 1904, helge von koch a trouvé une courbe continue et non dérivable. Onconstruit cette courbe, dite flocon ou île de von koch, par récurrence. http://www.ifrance.com/nobug/nobug1/article1/fract1/fractp14.htm

Les Fractales Intro: Des dimensions fractionnaires: - log(2)/log(3)=~0,63 pour la construction triadique de cantor - log(4)/log(3)=~1,26 pour le flocon de von Koch - log(3)/log(2)=~1.58 pour le triangle de Sierpinski Le "Zoo" des courbes fractales: - Le "flocon" de Von Koch: - La courbe de Peano: - La courbe du dragon: - Le tapis de Sierpinski: by NoRSfall Maintenant un petit peu de pratique: Programme pour dessiner un tapis de Sierpinski en C: L'ensemble de Mandelbrot - Les Biomorphes - Les IFS Download: Cliquez ici pour downloader cet article. Bibliographie: - Fractals Everywhere par Michael F. Barnsley - Les Objects Fractals: forme, hasard et dimensions par B. Mandelbrot

65. El Conjunto De Koch Translate this page Definidas por helge von koch en 1904, estas curvas se forman a partir de un segmento,por la sustitución de su tercio central por dos segmentos de longitud http://platea.pntic.mec.es/~mzapata/tutor_ma/fractal/koch1.htm

Práctica 2.- CURVAS POLIGONALES DE KOCH. Definidas por Helge von Koch en 1904, estas curvas se forman a partir de un segmento, por la sustitución de su tercio central por dos segmentos de longitud tambien un tercio, pero formando ángulos de 60º. Proceso que se repite recursivamente en cada segmento de las figuras que progresivamente se van obteniendo. Por tanto la poligonal de nivel 1 es un segmento: Para NIVEL=1 Para NIVEL=2 Para NIVEL=3 Para NIVEL=4 Para NIVEL=5 Para NIVEL=6 ACTIVIDAD A REALIZAR Elaborar los procedimientos LOGO para representar la Poligonal de Koch para un nivel n y una longitud dados.

66. Koch Close Window Tomorrow is Niels Fabian helge von koch's Birthday! We thank you foryour snowflakes. Happy Birthday koch. Born January 25 in Stockholm, Sweden. http://curvebank.calstatela.edu/birthdayindex/jan/jan24koch/jan25koch.htm

Close Window Tomorrow is Niels Fabian Helge von Koch's Birthday! We thank you for your snowflakes. Happy Birthday Koch Born: January 25 in Stockholm, Sweden Died: March 11, 1924 near Stockholm, Sweden

67. Die Fraktale Geometrie Ist Jener Junge Teil Der Mathematik Translate this page Bekannte Mathematiker wie Georg Cantor (1872), Guiseppe Peano (1890), David Hilbert(1891), helge von koch (1904), Waclaw Sierpinski (1916) oder Gaston Julia http://www.kuehnert.de/mg/ifs/frac004.htm

Klassische Fraktale Die sogenannten klassischen Fraktale wurden zu Anfang des vorigen Jahrhunderts veröffentlicht. Bekannte Mathematiker wie Georg Cantor (1872), Guiseppe Peano (1890), David Hilbert (1891), Helge von Koch (1904), Waclaw Sierpinski (1916) oder Gaston Julia (1918) - um nur einige zu nennen - entdeckten eine Anzahl von Kurven und Mengen mit - für die damalige Riege der Mathematiker - eigenartigen und geradezu anomalen Eigenschaften. Diese 'mathematischen Monster' oder 'Monsterkurven', wie man diese Kurven zu nennen pflegte, wurden im 19. Jahrhundert zu Kuriositäten und sonderbaren Anomalien deklariert, die eine Ausnahme und nicht die Regel darstellen sollten, und gerieten bis Mitte der 70ger Jahre fast in Vergessenheit. Darstellung klassischer Fraktale Die überwiegende Zahl der klassischen Fraktale entsteht durch einem iterativen Prozeß, indem bestimmte Elemente (z.B. gewisse Linienstücke oder Flächenteile) einem Ausgangsobjekt (= Initiator) hinzufügt oder entfernt werden. Ein Beispiel ist die nach Helge von Koch benannte Koch-Kurve.

68. Koch's Curve efficient implementation. koch's Curve. An example of a simple fractalimage is koch's curve, named after Swedish helge von koch. The idea http://www.cs.auc.dk/~normark/eciu-recursion/html/recit-note-koch.html

Page 14 : 41 Recursion * Introduction Introduction About the authors * Everyday recursion Everyday recursion Recursive visual effects Physical structures: A staircase Biological processes: Strawberry plants (1) Biological processes: Strawberry plants (2) 'Recursion' in dictionaries * Fractals and Curves Fractals Cantor's set Koch's Curve Fractal Essentials Fractals in a cultural perspective Settlement of Bamileke The Ba-ila settlement Hilbert Curves (1) Hilbert Curves (2) Building Hilbert Curves of order 1 Building Hilbert Curves of order 2 Building Hilbert Curves of order 3 Building Hilbert Curves of order 4 A program making Hilbert Curves * Recursive algorithms and recursive processes The Eight Queens Problem (1) The Eight Queens Problem (2) Backtracking and the Eight Queens Problem A solution to the Eight Queens Problem * Recursive data types Recursive datatypes Linear lists Binary trees * The implementation of recursion The implementation of recursive functions Implementation of recursion Stack development of normal recursion An alternative recursive definition of fak Stack development of the alternative fak A memory efficient implementation Koch's Curve An example of a simple fractal image is Koch's curve, named after Swedish Helge von Koch

69. Koch's Curve koch's Curve, An example of a simple fractal image iskoch's curve, named after Swedish helge von koch. http://www.cs.auc.dk/~normark/eciu-recursion/html/recit-slide-koch.html

Slide 14 : 41 Koch's Curve An example of a simple fractal image is Koch's curve, named after Swedish Helge von Koch

70. Koch Snowflake - Wikipedia segments connected end to end. It is a fractal and was brought to usin 1904 by helge von koch (a Swedish mathematician.). kochFlake.png. http://www.wikipedia.org/wiki/Koch_snowflake

Main Page Recent changes Edit this page Older versions Special pages Set my user preferences My watchlist Recently updated pages Upload image files Image list Registered users Site statistics Random article Orphaned articles Orphaned images Popular articles Most wanted articles Short articles Long articles Newly created articles Interlanguage links All pages by title Blocked IP addresses Maintenance page External book sources Printable version Talk Log in Help Koch snowflake From Wikipedia, the free encyclopedia. The Koch curve is one of the earliest fractal curves to have been described, appearing in a paper entitled "Une méthode géométrique élémentaire pour l'étude de certaines questions de la théorie des courbes plane" by the Swedish mathematician Helge von Koch You can imagine that it was created by starting with a line segment, then recursively altering each line segment as folows: divide the line segment into three segments of equal length. draw an equilateral triangle that has the middle segment from step one as its base. remove the line segment that is the base of the triangle from step 2.

71. Bate Byte 121 Junho/2002 - A Curva De Kock (Fractal Floco De Neve) Translate this page mostra um fractal conhecido como a Curva de koch, ou fractal do Floco de Neve, quefoi pesquisado inicialmente pelo matemático sueco helge von koch em 1904, e http://www.pr.gov.br/celepar/celepar/batebyte/edicoes/2002/bb121/curva.htm

A Curva de Koch (Fractal Floco de Neve) Autor Sergio Luiz Marques Filho Dando continuidade aos artigos sobre fractais, o desenho abaixo mostra um fractal conhecido como a Curva de Koch, ou fractal do Floco de Neve, que foi pesquisado inicialmente pelo matemático sueco Helge von Koch em 1904, e recebeu este nome por sua semelhança com um floco de neve. O fractal do floco de neve é uma excelente figura para entendermos os conceitos de fractais, pois o mesmo apresenta as características de fractais que vimos: Ao navegarmos na escala do fractal, e se tomarmos uma parte da figura ela parecer-se-á com qualquer outra parte do fractal; A cada iteração o perímetro do fractal aumenta, e, após n iterações, o mesmo tende ao infinito. O Fractal do Floco de Neve O fractal do Floco de Neve, consiste em um triângulo equilátero inicial, de onde tomamos cada um de seus lados e o dividimos em três segmentos de reta iguais. Retiramos, então, o segmento do meio e o substituímos por outro triângulo equilátero sem a base. Demonstramos abaixo as iterações de um dos lados do triângulo inicial. Iterações: Cálculo dos pontos Situação inicial: Após a primeira iteração.

72. Les Fonctions Continues Sans Derivees Translate this page C'est le suédois helge von koch qui donne le premier un exemple detelle courbe. Cette courbe est maintenant devenue classique. http://perso-info.enst-bretagne.fr/~brouty/Maths/noderiv.html

Retour Plan Il existe une version latex de cet article. Introduction Les travaux de Weierstrass Les travaux de Dini (1854 - 1918) Exemples ... Bibliographie Introduction fonction "...... On appelle fonction de x ou en général d'une quantité quelconque, une quantité algébrique composée de tant de termes que l'on voudra et dans laquelle x se trouve de manière quelconque, mélée ou non avec des constantes.Ainsi sont des fonctions de x." Les travaux de Weierstrass Pour F x a ab au voisinage de 0. posons: avec h et Majoration de On utilise la formule des accroissements finis: Si on suppose ab On remarque que si ab F x Minoration de avec et h avec et donc on a donc: Le nombre a est impair, donc est aussi impair et nous montrent que , par suite mais puisque et Ainsi , mais et Suposons que l'on ait: ce qui donne soit et mais comme nous avons choisi Quand m tend vers l'infini h tend vers car et l'expression tend vers l'infini car ab Les travaux de Dini (1854 - 1918) a b ] que nous noterons Les conditions pour que F x a b ] sont les suivantes: est uniformenent convergente.

73. Vorl_ws_0102 Translate this page koch-Schneeflocke Mathematisches Fraktal (helge von koch, 1904), Iterative Berechnungenvon Umfang und Fläche, Grenzübergang für unendlich kleine Skalen http://oie.mpg.uni-rostock.de/people/reinhard.dir/vorl_ws_0102/

Nichtlineare Dynamische Systeme Lehrveranstaltung (2 SWS V) im Rahmen des Graduiertenkollegs Stark korrelierte Vielteilchensysteme Montag 15.15 bis 16.45 Uhr, Seminarraum Wintersemester 2001/2002 Zum Inhalt der Lehrveranstaltungen: (15.10.2001, Mitschke/Mahnke) Die nichtlineare Physik (29.10.2001, Mahnke) Klassifikation dynamischer Systeme Zustandsraum, Zustandsvektor, Bewegungsgleichung, Trajektorie; Vergleich dynamischer Systeme nach verschiedenen Kriterien wie deterministisch/stochastisch; kontinuierlich/diskret; konservativ/dissipativ u. a. (05.11.2001, Mitschke) Superpositionsprinzip Bei linearen Systemen gilt das Superpositionsprinzip, bei nichtlinearen Systemen nicht. Fourierkomponenten orthogonal Dotierter Halbleiter, nichtlineare (logarithmische) Diodenkennlinie Elektronik: Mechanik: Federpendel Atomphysik / Optik: Licht-Materie-Wechselwirkung, Zweiniveau-Atom, Resonanz, Energieniveauschema mit Absorption, spontaner und stimulierter Emission. (12.11.2001, Mitschke) Fortsetzung: Atomares Zweiniveausystem in Wechselwirkung mit Lichtfeld als elementarste Wechselwirkung von ,,Signal`` mit ,,Material``. Absorption, Emission, spontane und induzierte Prozesse. Einsteinkoeffizienten. Ratengleichungsmodell. optischen Kerreffekt Unterscheidung Mitkopplung - Gegenkopplung Gegenkopplung Mitkopplung Nichtlineare dynamische Systeme in Physik und Nicht-Physik (19.11.01, Mahnke)

74. Biography-center - Letter V von koch, helge wwwhistory.mcs.st-and.ac.uk/~history/Mathematicians/koch.html;von Leibniz, Gottfried www-history.mcs.st-and.ac.uk/~history/Mathematicians http://www.biography-center.com/v.html

Visit a random biography ! Any language Arabic Bulgarian Catalan Chinese (Simplified) Chinese (Traditional) Croatian Czech Danish Dutch English Estonian Finnish French German Greek Hebrew Hungarian Icelandic Indonesian Italian Japanese Korean Latvian Lithuanian Norwegian Polish Portuguese Romanian Russian Serbian Slovak Slovenian Spanish Swedish Turkish V 192 biographies Vaca, Alva Nuñez Cabeza de www.pbs.org/weta/thewest/people/a_c/cabezadevaca.htm Vaca, Alva Nuñez Cabeza de www.newadvent.org/cathen/03126c.htm Vacca, Giovanni www-history.mcs.st-and.ac.uk/~history/Mathematicians/Vacca.html Vaga, Perino del www.getty.edu/art/collections/bio/a952-1.html Vailati, Giovanni www-history.mcs.st-and.ac.uk/~history/Mathematicians/Vailati.html Val, Patrick du www-history.mcs.st-and.ac.uk/~history/Mathematicians/Du_Val.html Vala, Katri www.kirjasto.sci.fi/kvala.htm VALEJO, Christiane www.christiane-valejo.com/Biography.htm Valentin, www.kirjasto.sci.fi/valentin.htm Valentin, Moise www.kfki.hu/~arthp/bio/v/valentin/biograph.html Valentine, www.knight.org/advent/cathen/15254b.htm Valerio, Luca

75. Enseigner Les Fractales Au Lycée Translate this page La courbe du mathématicien suédois helge von koch (1904) est un des premiersexemples historiques de courbe continue mais non dérivable en tout point. http://www2.ac-lille.fr/math/fractales.htm

76. DIMENSIONS OF THE FRACTALS One of perhaps the most famous fractals is koch's curve named after helge vonkoch, 1904. Find a dimension of the snowflake curve (of helge von koch). http://rc.fmf.uni-lj.si/matija/logarithm/worksheets/fractal.htm

DIMENSIONS OF THE FRACTALS Between the late 1950s and early 1970s Benoit Mandelbrot evolved a new type of mathematics, capable of describing and analysing the structured irregularity of the natural world, and coined a name for the new geometric forms: fractals . Fractals are forms with detailed structure on every scale of magnification. The simplest fractals are self-similar. Small pieces of them are identical to the whole. We are going to see only some very simple examples. Some pictures: The dimension of the fractal is very interesting. We are used to the idea, that a line is one-dimensional, a plane two-dimensional, a solid three-dimensional. But in the world of fractals, dimension aquires a broader meaning, and need not be a whole number. We are going to study the dimensions of the fractals on the example of Sierpinski gasket. This is obtained by repeatedly deleting the middle quarter of a triangle, removing smaller and smaller pieces, forever. The Sierpinski gasket can be thought of as being composed of three identical gaskets, each

77. DeutschesFachbuch.de Translate this page Buchcover Werner koch, Stuttgart † Aktualisierte Gehölzwerttabellen Bäume undSträucher als 3. Auflage, bearbeitet von helge Breloer - Auszug - Titelblatt http://abcatalog.net/buchtipp/landwirtschaft/3884876341.html

78. Perfect & Pathological Math topics from week one to von koch snowflake · General properties of logarithms· Biography Georg Cantor, helge von koch, Waclaw Sierpinski, David Hilbert. http://www.moscholars.org/curriculum/Perfect and Pathological Mathematics.htm

Teachers Perfect and Pathological Mathematics I. Course description Quick! When is a coffee cup equivalent to a donut? Is it possible for a shape to have infinite surface area and finite area? How do you know? In this class, we will meet the fringe elements of the world of mathematics: we'll encounter well-behaved and mathematically beautiful ideas and theorems, and we'll spend a lot of time with the misbehaving miscreants that have stymied long-held mathematical assumptions. We'll not only study the functions, curves and ideas that have reassured and rocked the world of math; we'll also study the means by which a theorem, proposition or lemma becomes mathematically valid. In addition, we'll explore the lives of the movers and shakers of the history of math and develop some ideas about the evolving nature of mathematics. Was it invented or discovered? What are the most pressing mathematical questions of our time? II. Instructor's educational preparation and current employment III. Rationale for inclusion in a program for gifted students

79. L-Systems - Koch Curve With each further step it increases its length by about 4/3. It firstappeared in a paper from Niels Fabian helge von koch in 1906. http://ai.toastbrot.ch/life/koch.php

backward main forward Koch Curves 1. Simple Koch Curve: The Koch Curves are very good samples of exact self-similarity at all scales after an "infinite" number of scales. After many iterations they already look like a final fractal, because of the finite resolution of your screen. The Koch Curve is of infinite length. With each further step it increases its length by about 4/3. It first appeared in a paper from Niels Fabian Helge von Koch in 1906. We start with a single straight line. By deviding the line into three equal parts and replacing the center part with two line segments each one-third in length we get...(click on the image below). The figure now contains four equal linesegments. In the next step each of the four segments is replaced again in the same way. This is to be repeated endlessly, so to generate the Koch Curve. F F+FF+F F+FF+F+F+FF+FF+FF+F+F+FF+F 2. Koch Snowflake: If we fit three Koch Curves together we get the Koch-Snowflake Recursions: 1-6 Angle : 60° Axiom : +FFF Rule : F=F+FF+F We start with a samesided triangle. Each side is a straight line.

80. Mathenomicon.net : Reference : Koch Curve generator. is usually taken to be an equilateral triangle. EtmyologyEnglish helge von koch Swedish mathematician (18701924). http://www.cenius.net/refer/display.php?ArticleID=kochcurve

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