81. 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

82. KaiRo Science Corner - Physik - Fraktale helge von koch 1904 entdeckte,eröffnet gute Möglichkeiten zur Beschreibung der fraktalen Eigenschaften http://www.kairo.at/science/physics/fraktalenatur.html

Die fraktale Geometrie der Natur Das Wort Fraktal =1,333; im zweiten Fraktale Dimension am Beispiel der Peano-Kurve fraktale Dimension War diese Kurve jetzt ein- oder zweidimensional? Mit unserem normalen ("topologischen") Dimensionsbegriff ist das nicht mehr nachvollziehbar. Mandelbrot nahm zur Beschreibung solcher Figuren denjenigen Begriff zu Hilfe, der in der Mathematik nach seinen Erfindern als Hausdorff-Besicovitch-Dimension oder einfach als gebrochene Dimension bekannt ist, und benannte ihn auch als fraktale Dimension. s den Skalierungsfaktor (hier 1/3), N Bei der Peano-Kurve wird die Initiator-Linie ebenfalls in drei Teile geteilt ( s =1/3) und durch den Generator ersetzt, der 9 solcher kleineren Linien ( N Welche fraktale Dimension besitzt dann die Koch-Kurve? Hier wird wiederum der Initiator mit dem Faktor s =1/3 geteilt, diesmal besitzt der Generator aber vier Teilstrecken N , also . Damit liegt sie, wie erwartet, zwischen der einer Linie ( D= 1) und der einer Ebene ( D Andere bekannte Fraktale: Sierpinkski-Dreieck und Cantor-Menge (aus einem Dreieck entstehen bei einer Skalierung s =1/2 drei Dreiecke) ergibt: . Damit erkennt man deutlich den Charakter der Cantor-Menge: Sie ist eine Punktmenge, die also zwischen der Dimension eines Punktes (

83. 04 Fraktaler Exkurs Translate this page Teilen der koch-Schneeflocke, besonders schön demonstriert (1904 in die Mathematikeingeführt durch den schwedischen Mathematiker helge von koch). http://www.mythen-der-buchkultur.de/Texte/04_Geschichte/Fliesstext/Fraktaler_Exk

04 Geschichte 3D und diachrone Kommunikationswissenschaft Fraktaler Exkurs Während die jedem vertraute euklidische Geometrie lediglich die Regeln zur Konstruktion von Objekten im 1-, 2- oder 3- (also ganzzahlig-) dimensionalen Raum kennt (z. B. von Punkt, Gerade, Quadrat, Kreis, Würfel, Pyramide u. a.), liefert die fraktale Geometrie eine Sprache zur Beschreibung komplexer Formen mit gebrochenen (lat. fractum) Dimensionen, der Fraktale. Mit ihrer Hilfe lassen sich die Ordnungsprinzipien in vermeintlich chaotischen Formen und Strukturen zeigen, z. B. das Prinzip der Selbstähnlichkeit : Ausschnitte einer Struktur gleichen ihr selbst: ein Ast ähnelt dem Baum, ein Zweig dem Ast, die Veräderung im Blatt dem Zweig. Eine Vielzahl natürlicher Formen hat deshalb fraktale Eigenschaften - ganz im Unterschied zu geometrisch idealisierten Gebilden wie einer Kugel, die in der Natur nicht vorkommt. Anders als die Begrenzungslinien eines Dreiecks sind die Ränder fraktaler Strukturen nicht glatt, sondern unendlich rauh - jede Vergrößerung zeigt neue Details. Natürliche Strukturen wie Wolken, Gebirge, Küsten- oder tektonische Bruchlinien, Blutgefäßsysteme, Pflanzenformen oder mineralische Oberflächen sind scheinbar unüberschaubar komplex, besitzen tatsächlich aber eine geometrische Regelmäßigkeit, die sogenannte Skaleninvarianz : Unabhängig von der Auflösung/Skalierung findet man immer wieder die strukturtypische Grundform.

84. 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.

85. 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.

86. 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.

87. Methode Koch Methode koch. "ES KOMMT ÜBERHAUPT NICHT AUF DAS IM BERUF ERREICHTE AN, SONDERN AUSSCHLIESSLICH AUF DAS DURCH WERNER koch (23.4.1927 28.4.1993). helge Breloer. Meppener Straße http://www.methodekoch.de/

Methode Koch "ES KOMMT ÜBERHAUPT NICHT AUF DAS IM BERUF ERREICHTE AN, SONDERN AUSSCHLIESSLICH AUF DAS DURCH UNSERE TÄTIGKEIT GELERNTE UND GEWORDENE. ABER DIESER BERUF BIETET EINE AUSGEZEICHNETE MÖGLICHKEIT, DAS ZU LERNEN, WAS WICHTIG IST, GOTT UND DEN MENSCHEN ZU DIENEN." WERNER KOCH (23.4.1927 - 28.4.1993) Helge Breloer 49733 Haren (Ems) Tel.: 05932-6490 Fax: 05932-2174 www.baeumeundrecht.de email: info@methodekoch.de Sie sind Besucher Nr.

88. Von Koch Translate this page Questa curva limite è stata essenzialmente proposta dal matematico tedesco Helgevon koch nel 1904 ed ha un'altra interessante proprietà non ammette http://digilander.libero.it/lucianobattaia/matematica/a_fiocchineve/pg1.htm

Fiocchi di neve - Indice "Un filo sottilissimo comunque disposto su di un piano, il segno tracciato dalla punta di una matita che si fa scorrere su di un foglio, il contorno di una superficie piana, ci danno l'idea di ciò che chiamiamo linea piana". Si tratta dell'introduzione al concetto di curva preso da un comune testo di geometria. É in effetti in questi termini che abitualmente pensiamo ad una curva. In realtà il concetto di curva è molto complesso e qui vogliamo far vedere su qualche esempio come le "definizioni" sopra riportate vadano "prese con le pinze". (Per comprendere appieno quanto diremo è opportuno fare riferimento al noto processo che porta alla rettificazione della circonferenza Esaminiamo ora il triangolo equilatero inscritto in un circonferenza, e supponiamo, per semplicità, che esso abbia lato 1. (Basterà quindi che il raggio della circonferenza sia Se dividiamo ciascuno dei tre lati in tre parti uguali, togliamo la parte centrale e la sostituiamo con i due lati di un triangolo equilatero di lato , otteniamo una figura come quella qui a fianco riportata, comprendente dodici lati tutti di lunghezza Ogni lato del triangolo equilatero di partenza è sostituito da quattro segmenti di lunghezza : la figura avrà come contorno una spezzata di 3 4 lati, con un perimetro lungo

89. Problem Set 1 Among the most famous line systems are the von koch snowflake, first described byHelge von koch in 1904, the Peano curve, the Hilbert curve, and the Cantor set http://www.cs.dartmouth.edu/~brd/Teaching/AI/Homeworks/ps1.html

Problem Set 1 Issued : Monday, January 5 Due : Friday, January 16 For help on this problem set, Read the handouts and the notes on the course homepage Come to Recitation Sections , or send email to: Susanna Leng, Undergrad TA Mintcho Petkov, Undergrad TA Sanjay Khanna, Graduate TA 223 Sudikoff; tel: 6-1639 (W); 603-448-6306 (H) Alik S. Widge, Undergrad TA Reading assignment for this problem set: Read the handouts on Dylan. Before you do anything, please read about our homework policy carefully. It contains a wealth of good advice that can save you a lot of headaches later on. There are also rules about collaboration and when, where and how to pass in homework. Please read and follow, under penalty of extreme disfavor. Output We will require output for each problem in each problem set, unless otherwise noted. To print output, simply copy it from the NOODLLE window and paste into an editor. Many students find it useful to put their code and output into the same file. That's fine. Remember that the output you give must be that produced by your code. Anything else is a violation of academic integrity and is cheating. Even if the output you give shows that the function does not work, you will receive full credit for output.

90. Johanneum Lüneburg Koch-Fraktal Translate this page Sie kann als Symbol für diese ganze Fraktalgattung gelten. Der MathematikerHelge von koch hat sie zu Beginn dieses Jahrhunderts vorgestellt. http://rzserv2.fh-lueneburg.de/u1/gym03/homepage/faecher/mathe/chaos/linde/koch.

Chaos und Fraktale Informationssystem Mathematik Chaos Wegfraktale Galerie Die Kochkurve, ein ganz besonderes Wegfraktal und L-System Kochkurve stufenweise Dimension LOGO Galerie der Wegfraktale ... Leitseite der Wegfraktale Generator ist der gerade Strich. Initiator ist eine 60°-Zacke. mathematische Fraktal , die Kochkurve, ist die Grenzfigur dieses Prozesses. Man kann das Fraktal denken , sehen kann immer nur Vorstufen. Die Kochkurve ist streng Damit ist die der Kochkurve d = log(z) / log(k) = log 4 / log 3 = 1,26. Man kann sie auch experimentell bestimmen durch Messung der Boxdimension Drei Exemplare der Kochkurve im gleichseitigen Dreieck angeordnet ergeben die kochsche Schneeflockenkurve . Sie ist ein auf endlichem Platz eine unendlich lange Kurve untergebracht wird. Realisierung von Hand: tun Realisierung mit rekursiven Prozeduren in LOGO PR koch :n :breite wenn :n =0 dann vw :breite rk koch :n-1 :breite / 3 li 60 koch :n-1 :breite / 3 re 120 koch :n-1 :breite / 3 li 60 koch :n-1 :breite / 3 ENDE Die Schneeflocke wird dann so verwirklicht: bild re 30 koch 4 150 re 120 koch 4 150 re 120 koch 4 150 Realisierung mit rekursiver Turtlegraphik-Prozedur in Pascal Leitseite Realisierung mit Lindenmayer-Systemen Kochkurve Das Axiom F (Stufe 0).

91. CS312 FA00 Problem Set 1 Among the most famous fractals are the von koch snowflake, first described by Helgevon koch in 1904, the Peano curve, the Hilbert curve, and the Cantor set. http://www.cs.cornell.edu/Courses/cs312/2000fa/homework/ps1/ps1.html

CS312 Fall 2000 Problem Set 1 Fractals Issued : Tuesday, September 5 Due : Tuesday, September 12 in class Any changes to the assignment will be marked like this The template for your code is available as ps1.sml (for Win32) or ps1-gs.sml (for Unix). Please put your code within the template as indicated to make grading easier. Before you do anything, please read about the entire assignment carefully . It contains a wealth of good advice that can save you (and us) a lot of headaches later on. For this assignment, you are to work alone . Please turn in a copy of your code in class or in Upson 303 before 4pm. No late assignments will be accepted, so please start early. There will also be a web submission form for the code announced on the course home page. Watch the course home page for announcements regarding this homework. If you have trouble with the course software or are generally stuck, come to our office hours or consulting hours . Alternatively, post a question to the newsgroup cornell.class.cs312 or email us at cs312@cs.cornell.edu

92. área Fractal - Koch & Sierpinski koch y Sierpinski. En 1.904 NielsHelge von koch (1870-1924) define la curva que lleva su nombre. Se forma (fig. http://www.arrakis.es/~sysifus/kochsier.html

Curvas de Koch y Sierpinski 1, 4/3, 16/9, 64/27, 256/81... , L=(4/3)^k 1, 3/4, 9/16, 27/64, 81/256... , A=(3/4)^k Variaciones Figura 1 Figura 2 Figura 3 Figura 4 Figura 5 Figura 6 Figura 7 index intro software misc

93. Historical Notes: History Of Fractals Later came geometrical figures example (c) on page 191 was introduced by Helgevon koch in 1906, the example on page 187 by Waclaw Sierpinski in 1916 http://www.wolframscience.com/reference/notes/934a

From: Stephen Wolfram, A New Kind of Science Notes for Chapter 5: Two Dimensions and Beyond Section: Substitution Systems and Fractals Page 934 History of fractals. The idea of using nested 2D shapes in art probably goes back to antiquity; some examples were shown on page 43. In mathematics, nested shapes began to be used at the end of the 1800s, mainly as counterexamples to ideas about continuity that had grown out of work on calculus. The first examples were graphs of functions: the curve on page 920 was discussed by Bernhard Riemann in 1861 and by Karl Weierstrass in 1872. Later came geometrical figures: example (c) on page 191 was introduced by Helge von Koch in 1906, the example on page 187 by Waclaw Sierpinski in 1916, examples (a) and (c) on page 188 by Karl Menger in 1926 and the example on page 190 by Paul Lévy in 1937. Similar figures were also produced independently in the 1960s in the course of early experiments with computer graphics, primarily at MIT. From the point of view of mathematics, however, nested shapes tended to be viewed as rare and pathological examples, of no general significance. But the crucial idea that was developed by Benoit Mandelbrot in the late 1960s and early 1970s was that in fact nested shapes can be identified in a great many natural systems and in several branches of mathematics. Using early raster-based computer display technology, Mandelbrot was able to produce striking pictures of what he called fractals. And following the publication of Mandelbrot’s 1975 book, interest in fractals increased rapidly. Quantitative comparisons of pure power laws implied by the simplest fractals with observations of natural systems have had somewhat mixed success, leading to the introduction of multifractals with more parameters, but Mandelbrot’s general idea of the importance of fractals is now well established in both science and mathematics.

94. Fractals Part 1 Translate this page Les mathématiciens du début du XXe siècle (Georg Cantor, Felix Hausdorff ou Helgevon koch), qui s'interrogeaient sur la notion de dérivabilité, avaient http://olivier.sc.free.fr/logosc/fractalo/fract000.html

"To iterate is human, to recurse is divine" fractal,ale,als (adjectif) (Toute La Connaissance) http://www.tlc-edusoft.fr FRACTALES frangere Bernard Pire http://www.encyclopaedia-universalis.fr Bibliographie En plus du GODEL, ESHER, BACH et du Visual Art, Mathematics and Computers bibliographie de la Compilation Logo , je vous propose les ouvrages suivants : H. PEITGEN, The Art of Fractals, a Computer Graphical Introduction , Springer-Verlag, Berlin, 1988 J. F. GOUYET, Physique et structures fractales, Masson, Paris, 1992 B. MANDELBROT, Les Objets fractals : forme, hasard et dimension, coll. Champs, Flammarion, Paris, 1995 Quelques noms Felix Hausdorff (1868-1942) ; Helge von Koch ou/et Jacques Mandelbrojt Surfer un peu Ne pas oublier les autres images fractales page ici A ne pas manquer : Vous pouvez aussi visiter ces Sites : http://www.eclectasy.com/Iterations-et-Flarium24/ http://fractals.iuta.u-bordeaux.fr/ http://raphaello.univ-fcomte.fr/IG/Fractales/Fractales.htm Le fractal de Rauzy : http://www.irisa.fr/symbiose/people/siegel/Pro/page_dessins.html

95. Version Professeur -- Page VINET Translate this page pour obtenir des explications plus complètes. Introduction. HelgeVon koch mathématicien suédois, 1870-1924. Il est le premier, en http://www.aromath.net/Page.php?IDP=292&IDD=0

96. Philosophy And Computers --- Links - Fractals It has also java applets, which draw Sierpinski Triangle and koch Snowflake. koch'sSnowflake Web page with a java applet, which draws koch's Snowflake. http://www.ic.sunysb.edu/Class/phi365/fractals.html

Fractals Tutorials Math of Fractals Fractal Galleries Programming ... Complex Numbers Tutorials Tutorials Fractal Lesson by Cynthia Lanius - Mathematics lessons of fractals. This website has answer on frequently asked questions about fractals. It has also java applets, which draw Sierpinski Triangle and Koch Snowflake. Fractal Tutorials - Website with three separate tutorials, each with a different level of difficulty. Infinity and the Area of a Sierpinski Gasket Chaffey's Fractal Links on the Web Organized website with tutorials and links to other resources on the Internet. The Fractory - An Interactive Tool for Creating and Exploring Fractals The Discovery of Fractals - This site talks about the history of fractals. Mandelbrot Set Explained - Explonation of the Mandelbrot Set for people who don't want to hear about numbers. Biographies Biography of Helge von Koch (1870-1924) Biography of Benoit Mandelbrot (1924-0BC) Koch Snowflake with Maple Recursion and making a Koch Snowflake with Maple.

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