41. Helge Von Koch Université Montpellier II Translate this page helge von koch (1870-1924). Cette image et la biographie complète en anglaisrésident sur le site de l’université de St Andrews Écosse http://ens.math.univ-montp2.fr/SPIP/article.php3?id_article=1388

42. TPE Fractales : Le Flocon De Von Koch Translate this page Pour la courbe de von koch (du nom du mathématicien helge von koch), on remplacele tiers central de chaque segment par un triangle équilatéral sans base http://lekernel.lya-fr.com/tpefractales/vonkoch.php

@import url("style.css"); BOURDEAUDUCQ Sébastien / RIQUET Jean Charles TPE Fractales Vous êtes ici : Version imprimable Sommaire Page d'accueil et introduction I - Présentation Définition d'une fractale Le flocon de Von Koch Le triangle de Sierpinski L'ensemble de Mandelbrot Autres fractales basées sur les complexes La dimension fractale II - Les fractales dans la nature 1. Etude d'objets fractals naturels La côte de Bretagne Chez les végétaux : Le chou-fleur Le chou romanesco Les fougères Dans le corps humain : L'intestin grêle Les poumons Le réseau coronarien 2. La modélisation des fractales naturelles Les L-systèmes IFS Conclusion Divers Biographies des personnes célèbres ayant étudié les fractales Benoît Mandelbrot Gastion Julia Waclaw Sierpinski Helge Von Koch Michael Barnsley Annexes Bibliographie Le TPE Nous contacter Livre d'or E-mail Le flocon de Von Koch D'après la définition, la méthode la plus simple pour obtenir une courbe fractale va être de partir d'une figure géométrique (appellée l' initiateur ) puis de remplacer une de ses parties par une autre figure, le

43. Niels Fabian Helge Von Koch Von Koch Nato Il 25 Gennaio 1870 A http://alpha01.dm.unito.it/personalpages/cerruti/Az1/koch.html

Niels Fabian Helge von Koch Nato il 25 gennaio 1870 a Stoccolma, morto l'11 marzo 1924 a Stoccolma. Fu studente di Mittag-Leffler e gli succedette nel 1911 all'Università di Stoccolma. E' famoso per la curva di Koch, costruita dividendo una linea in tre parti uguali e sostituendo il segmento intermedio con gli altri due lati del triangolo equilatero costruito su di esso. Questa costruzione si ripete su ognuno dei segmenti (ora 4) e così all'infinito. Si ottiene una curva continua di lunghezza infinita e non derivabile in alcun punto. I principali risultati di Koch riguardano i sistemi di infinite equazioni lineari in infinite incognite.

44. DIMENSIONS OF THE FRACTALS Find a dimension of the snowflake curve (of helge von koch). You will draw itwith plotting simplifed expression koch_CURVE(2,-2,2,2,n), where n can be 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

45. Making Order Out Of Chaos: Iteration And The Kotch Curve The image we are left with is called the koch Curve, named after the famousmathematician helge von koch. The image has an infinate perimter, http://library.thinkquest.org/12170/theory/koch.html

Iteration and The Kotch Curve Let's propose a question: How long is the coast of England? Benoit Mandelbrot (see Benoit Mandelbrot in the History Section) would say that it is infinate. While this may seem absurd, let's explain ourselves by examining the Kotch Curve , one of the simplest types of fractals. First, draw an equilateral triangle (a triangle with three sides of equal length): Next, divide each side into three equal segments, and use the middle segment of each side as part of a new equaliteral triangle: The following applet shows the how a new triangle is attached to every side of the existing figure. The process can then be repeated on each new segment. (Repeating a mathematical process over and over is called iteration If we iterate infinately, we will be left with a snowflake-like image. Its edges are seemingly curves, but are really composed of an infinate amount of lines of infinately small length: The image we are left with is called the Koch Curve , named after the famous mathematician Helge von Koch. The image has an infinate perimter, yet a finate area (if the initial triangle were inscribed in a circle, the final Koch Curve would still be within the circle's boundaries). No matter how closely we enlarge a portion of the image, we will still see an infinately complex system of lines. The Kotch Curve can be applied to our English coast enigma. Suppose a person measures a portion of the coast and says that it is 1000 meters long. How accurate is this measurement? Suppose for a moment that our measurer, finding the cost to be 1000 meters long, only measured in a staight line. If someone were to measure the coast again, but this time carefully measure around every part of the land's extrusion into the sea, the distance would become considerably greater. We could then ask that the coastline be measured around each rock, pebble, grain of sand, molecule, atom . . . Each time we measure on a smaller scale, the distance of the coast becomes greater. Theoretically, if we could measure the coast with infinate precision, its length would be infinitely long.

46. MathBlues.com > Featured Articles > Dazzling Fractals One of the mathematicians he researched was helge von koch. In 1904, almost 50years earlier, koch had a brush with fractals. koch constructed an odd figure http://www.mathblues.com/mainpages/sampleissue/articles/1/

SAT Tips Poll Math Doodles Contests ... Sample Issue Clouds are not spheres, mountains are not cones, coastlines are not circles and bark is not smooth, nor does lightning travel in a straight line. - Benoit Mandelbrot I n nature, irregularity rules! Look around you and notice that natural forms don't have much respect for the naïve shapes of Euclidean geometry (squares, circles, triangles, spheres, cones, etc...). Straight lines and smooth curves, which are the key features of Euclidean geometry, are only rough approximations for describing irregular shapes. The irregularity in natural systems is a sign of their incredible complexity. How do you understand this complexity? In the 1970s, Yale professor Benoit Mandelbrot searched for a new geometry to describe irregularity. Mandelbrot's search led him to shapes that he named fractals. He coined the word fractal from the Latin fractua , which means irregular. He found that fractals offer a useful and more realistic representation, or model, of natural systems. Inspired by the work of several mathematicians before him, Mandelbrot channeled his creative genius into developing the new geometry.

47. Text4 In the early 1900s, the Swedish mathematician helge von koch constructed an objectmade up of a sequence of identical steps. He started with an equilateral http://www.scientific-religious.com/text4.html

IV. The Principle of Self-Similarity Euclidean geometry is the science that deals with regular one-dimensional lines, two-dimensional planes and three-dimensional solids. Nevertheless, about the same time the theory of self-organization was being developed but independently of it, the French mathematician Benoit Mandelbrot formulated a geometry that portrays the irregular shapes of clouds, mountains, coastlines, leaves, flowers, trees, and the countless other irregular shapes found in Nature. He named it fractal geometry. (The word fractal comes from the Latin "fractus" which means "broken." Fractus is also at the core of the words "fracture" and "fraction".) An example of an artificial fractal is the Koch curve. In the early 1900s, the Swedish mathematician Helge von Koch constructed an object made up of a sequence of identical steps. He started with an equilateral triangle. On each side he added equilateral triangles with sides half the size of the original, making a Star of David. The operation was repeated which led to a shape similar to a snowflake. Koch discovered that by continuing the process endlessly on an increasingly smaller scale it became a continuous curve containing an infinite number of minute bends. Through this process of repetition the triangle became a fractal. The Koch curve is of infinite length yet finite space; it is more than a line with one dimension but less than a plane with two dimensions; it is an endless loop where lines never cross; and at any level of magnification a portion resembles the whole. Because the repetitions are identical, the Koch curve is a fractal that is self-similar under linear transformation.

48. Introduction To Spaceblooms They settled on a modular design that was inspired by the work of helge von koch,after whom the garden was named. They called the spacebloom koulekouli. http://www.spacebloom.net/intro/

@import "../xs/spacebloom.css"; Skip navigation intro catalog appendix now download ... book a brief history fbloom prototype In September 2235, Matoni Wulffi, a 21-year-old student of engineering, entered his final year at the University of Adanac. As a graduation project, he chose to build a self-propagating, fragrance-producing synthetic flower capable of an auton-omous existence in deep space. The project was deemed to be too complex and Matoni was advised to simplify it. Undaunted by the apparent impossibility of the task, he set out to unite the disparate technologies needed to create the object of his dreams: the flowerbloom (fbloom). The integration of existing technologies proved to be his greatest challenge. Despite several attempts to standardize software, hardware and bioware communications over the years, the agreed guidelines were often too vague to ensure full compatibility and usually were implemented only when advantageous to the creators. Matoni solved the problem with a custom-engineered substance that joined the various hardware and bioware components together physically and facilitated communications as well. The substance was gluevins, and Matoni received a universe-wide patent, no. U1008-0021, for his invention shortly after graduation. The fbloom prototype consisted of four solpans, two alseco betteries, a conventional emag propulsion unit, and a UPU, all of which were mounted around a tubular frame that housed the centrepiece: the unimbler-based matter processor. The whole unit was controlled by a language Matoni created specifically for the project. It was called *TNG (star tongue).

49. Problem Set 1 are the von koch snowflake, first described by helge von koch in 1904, In the von koch snowflake, a level k+1 side consists of four level k sides. 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.

50. Manfred Boergens - Briefmarke Des Monats Januar 2004 von Niels Fabian helge von koch (1870 - 1924). Im Jahre1904, also vor 100 Jahren, konstruierte der Stockholmer Mathematikprofessor http://www.fh-friedberg.de/users/boergens/marken/briefmarke_04_01.htm

Mathematik auf Briefmarken # 37 Liste aller Briefmarken vorige Marke zur Leitseite Briefmarke des Monats Januar 2004 Schweden 2000 Michel 2207 / 2208 Fraktale "Schneeflocke" von Niels Fabian Helge von Koch (1870 - 1924) Im Jahre 1904, also vor 100 Jahren, konstruierte der Stockholmer Mathematikprofessor Helge von Koch fraktalen Kurve Konstruktion der Koch'schen Schneeflocke Die Koch'sche Schneeflocke ist eine fraktale Kurve haben. Dann ist die n -te Iteration ein Polygon mit n Seiten n Umfang n n a (a . Das Ausgangsdreieck ( . Bei der i -ten Iteration kommen i-1 i . In der i i-1 i i-1 i n (endliche geometrische Reihe) noch n n n n Fraktale Dimension Fraktalen Gebilden kann man eine fraktale Dimension D zuordnen. Die Koch'sche Schneeflocke hat die Dimension D log log c r c und r D log c log r Sierpinski-Dreieck Waclaw Sierpinski c und r r . So ergibt sich D log log n -ten Iterationsschritt in r n c r , also D n -ten Iterationsschritt in r c r , also D n -ten Iterationsschritt in r c r und D entstanden. L L Sei nun s L(s) s L(1) L(s) L aufzufassen, zudem mit der Vorstellung, dass lim L(s) L s L(s) s gegen Unendlich.

51. Arlo Caine's Web Page / Mathematics / Limits With ... Niels Fabian helge von koch (18701924) was a swedish mathematician who firstplayed with the figures we are discussing. He noticed that as the stages http://math.arizona.edu/~caine/chaos.html

Limits with von Koch's Curve, Sierpinski's Gasket, and the Chaos Game Patterns The natural world is awash with detail. Even in the bleak lanscape of a frozen tundra, patterns can be found at all scales, whether in the gentle curve of a wind swept serac or in the intricate crystalline structure of a snowflake. It is fortunate for us that many of the observable patterns in nature form as the result of a process. A step by step series of instructions to be carried out. Some processes are more complicated than others. Ice crystal formation is a fairly simple process when compared with the fluid dynamics relevant to the creation of clouds, for example. In science one tries to develop theoretical models to predict these patterns, thereby understanding the process, and then tests these predictions against experiement. Often these predictions use mathematics, and during the development of the full model, many interesting questions arise. An example is pictured schematically at right. In the picture at right, suppose for the sake of

52. Die Fraktale Koch-Kurve Als Java-Applet Translate this page Eigentlich wollte helge von koch 1904 nur eine Kurve beschreiben, die, obwohlsie fast überall stetig ist, an keiner Stelle eine Tangente besitzt, http://www.jjam.de/Java/Applets/Fraktale/Koch_Kurve.html

JJAM Home Applets Tetraeder ... Kugel 2 Fraktale: Juliamenge Juliamenge MA Julia-Generator Koch-Kurve ... Lindenmayer-System 2 Mathematik: Funktionsplotter Eratosthenes-Sieb Miller-Rabin-Test Verschiedenes: Morsezeichen-Ticker Analoguhr Scripts Kontakt - Applets : Fraktale : Koch-Kurve - Die fraktale Koch-Kurve als Java-Applet. Mehr Zacken mit linkem Mausklick - Weniger mit rechtem Mausklick. [Die fraktale Koch-Kurve als Java-Applet mit Quellcode zum Download. Das Applet der Koch-Kurve lässt sich allerdings nur mit aktiviertem Java betrachten !] Die Koch-Kurve (auch Schneeflockenkurve oder kochsche Insel). Nach Helge von Koch, schwed. Mathematiker, 1870-1924 KochKurve.java (Helge von Koch) Download Koch_Kurve.zip (Applet und Code ca. 2 kb) Impressum Datenschutz Nutzung eMail

53. Fractals In 1904 the mathematician helge von koch gave an exemple of a curve that doesn thave a tangent anywhere. For the mathematicians of that time a shocking http://mathsforeurope.digibel.be/Leenfrac.html

Fractals By Els Cant, Leen Gillis, Katrien Janssens, Laetitia Parmentier 1. Some historical background The groundwork for this subject was started in the early part of this century mainly by two French mathematicians, Gaston Julia(1893-1978) and Pierre Fatou. Julia, after whom some of these sets are named, was a soldier during World War I. During an offensif designed to celibrate the Kaiser's birthday, he was wounded and lost his nose. After that, he had to wear a leather mask. A great deal of work was done on this subject for several years, but later in the 1920s the study of this field almost died out. The subject was renewed in the late 1970s through the computer experiments of Dr. Benoit Mandelbrot(also French) at Yale University. In honor of Dr. Mandelbrot, one of the sets, which he explored, was named after him. Other mathematicians such as Douady, Hubbard, and Sullivan worked also on this subject exploring more of the mathematics than the applications. Since the late 1970s this subject has been at the forefront of contemporary mathematics. Two properties of a fractal: 1. The object is self-similar and chaotic, its also based on iteration.

54. Von Koch Curve Neils Fabian helge vonkoch (18701924) curve first appear in his paper Une méthode géométrique......Draws nth iteration of von koch curve. http://members.lycos.co.uk/ququqa2/fractals/Koch.html

von Koch curve Instructions: Draws n th iteration of von Koch curve. Description: Neils Fabian Helge von Koch (1870-1924) curve first appear in his paper Une méthode géométrique élémentaire pour l'étude de certaines questions de la théorie des courbes plane published in 1906. This curve is constructed by dividing a line segment into three equal parts and replacing the middle segment by the other two sides of an equilateral triangle constructed on the middle segment. Repeat on each of the (now 4) segments. Repeat indefinitely. It gives a continuous curve which is of infinite length and nowhere differentiable. You can observe process of creation of Koch curve by setting up the first step and incrementing number of steps using " " button. If one starts with an equilateral triangle and applies the construction, one gets the von Koch snowflake (sometimes called the von Koch star) as the limit of the construction. Others von Koch curves: Von Koch curve for a square Von Koch curve for triangles Random von Koch curve Contents ... Back to main page

55. The ORESME Home PageVon Koch Section 4 The 1906 reference is helge von koch, Une méthode géométrique élémentaire pourl étude de certaines questiones de la théorie des courbes planes, Acta http://www.nku.edu/~curtin/grenouille.html

Von Koch Section IV. Please Email comments or suggestions to: curtin@nku.edu You may view the four pages of the 1906 paper that do not appear in the 1904 version. Acta Mathematica (1906), 145-174. You have Vardi's translation of pp. 145-170. Page 171 Page 172 Page 173 Page 174 ... Dan Curtin's Home Page

56. ORESME NKU Sept 1998 tangents constructible from elementary geometry by helge von koch (an The question of what von koch meant by a continuous uniform function arose. http://www.nku.edu/~curtin/oresme_sep_98.html

Please Email comments or suggestions to: curtin@nku.edu or to: otero@xavier.xu.edu ORESME Home Page Dan Curtin's Home Page

57. The Cushman Network - Fractals David Hilbert (1891), helge von koch (1904), Waclaw Sierpinski(1916), Another wellknown fractal is the von koch curve, or the von koch snowflake. http://cushman.net/projects/fractals/

Fractals and Chaos in Nature In the past two decades, scientists and mathematicians have developed a new way of looking at the universe around us, a new science that better describes the irregular shaped objects we find in nature and math. As James Gleick put it, "This new science, called chaos, offers a way of seeing order and pattern where formerly only the random, erratic, the unpredictable - in short, the chaotic - had been observed". Scientists had come upon an important tool in understanding nature. This science, along with the closely related science of fractals, models real-world situations better than anything else before. In 1961 at MIT, Edward Lorenz developed a model for an ideal weather system with few variables. He came up with three equations to reflect the changes on a computerized graph. These equations are defined to be: dx/dt = 10(y-x) dy/dt = xz+28x-y dz/dt = xy-(8/3)z He was studying changes in the weather, but he unknowingly founded the science of chaos. He discovered that small changes in the initial conditions would produce large differences in the long run (Stevens 63) The computer that Lorenz ran his system of equations on would compute the digits out to an accuracy of six decimal places. When Lorenz wanted to re-simulate a section on the graph that was produced, he started the computer over again at the beginning of the section in question, only with three digits of accuracy, instead of six. After a short period of time, Lorenz could see a large difference in the two graphs. This led to the discovery of the Lorenz attractor, a butterfly-shaped graph. When these equations are graphed on a computer, the output is chaotic, but orderly. These equations model such natural phenomena as the flow of fluid, or the movement of a water wheel

58. Koch Snowflake This snowflake was created by Niels Fabian helge von koch, to demonstrate acurve which is continuous but NOWHERE differentiable (ie no matter how close http://wso.williams.edu/~nyates/fractal/kochsnowflake.html

The Koch Snowflake One of the most famous fractal curves of all time! This "snowflake" was created by Niels Fabian Helge von Koch, to demonstrate a curve which is continuous but NOWHERE differentiable (i.e. no matter how close you get, it is still connected to itself, but has sharp corners instead of a smooth curve). We begin with an equilateral triangle. Each side is split into three equal pieces, the middle piece is erased, and a bump (a baseless equilateral triangle) appears in its place. This same process is repeated ad infinitum, resulting in the Koch snowflake. A few of the steps are reproduced down below. The Koch snowflake has infinite perimeter (since it becomes 4/3 as long after each iteration), but finite area (if you circumscribed a circle about the original triangle, the end snowflake would fit in the same circle). This is similar to the Koch Quadric Island and to many other fractals. The Snowflake's fractal dimension is (log 4)/(log 3) Several steps in the iteration of the Koch Snowflake: Step 0, an equilateral triangle.

59. Motivate : Project Work 2 The koch curve. as the snowflake curve you ll soon see why - and it wasfirst invented by a Swedish mathematician called helge von koch in 1904. http://motivate.maths.org/conferences/conf19/c19_project2.shtml

@import url(../../motivate.css); /*IE and NN6x styles*/ Home Conferences Long projects Get involved! ... Publications ABOUT MOTIVATE about our conferences conference charges ... contact us CONFERENCE PROGRAMMES mathematics science cross-curricular Back to : Fractals Main Page Display maths using: fonts images Need help? The Koch curve All the triangles are equilateral in this part of the project. This is also known as the "snowflake curve" - you'll soon see why - and it was first invented by a Swedish mathematician called Helge von Koch in 1904. Everyone at the first videoconference will have done this section already. It's here so that other schools can use these materials, even if they have not taken part in the videoconferences. Take a large piece of sugar paper, and draw an equilateral triangle whose sides are each 27cm long. Step 1. Divide the triangle side length by 3. Step 2 . Make a triangle with this side length for each side of the previous triangle. How many triangles do you need? Record this number each time you go through these instructions. Step 3 . Stick one new triangle to the middle of each of the sides of the previous triangle. Repeat Steps 1 to 3 until you have made triangles that are 1cm side length, and stuck them on.

60. Fractals: Von Koch Curve Niels Fabian helge von koch Born 25 Jan 1870 in Stockholm, Sweden Died 11 March1924 in Danderyd, Stockholm, Sweden Niels Fabian helge von koch attended a http://users.swing.be/TGMSoft/curvevonkoch.htm

DisplayHeader( "Geometric Fractals", "The Von Koch Curve", 0, "main_fractals.htm", "Back to Fractals Main Page"); Content Introduction Construction Properties Variations Author Biography All pictures from WinCrv Introduction The Von Koch curves, named from the swedish mathematician Helge Von Koch who originally devised them in 1904, are perhaps the most beautiful fractal curves. These curves are amongst the most important objects used by Benoit Mandelbrot for his pioneering work on fractals. More than any other, the Von Koch curves allows numerous variations and have inspired many artists that produced amazing pieces of art. Construction The construction of the curve is fairly simple. A straight line is first divided into three equal segments. The middle segment is removed and replaced by two segments having the same length to generate an equilateral triangle. Applying such a 4-sides generator to a straight line leads to this: This process is then repeated for the 4 segments generated at the first iteration, leading to the following drawing in the second iteration of the building process: The third iteration already gives a nice picture: Increasing the iteration number provides more detailed drawings. However, above 8 iterations, the length of the segments becomes so small ( in fact, close to a single pixel) that further iterations are useless, only increasing the time of curve drawing.

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