21. Math Lessons - Helge Von Koch Math Lessons helge von koch. His father, Richert Vogt von koch (1838-1913)was a Lieutenant-Colonel in the Royal Horse Guards of Sweden. http://www.mathdaily.com/lessons/Helge_von_Koch

Search Mathematics Encyclopedia and Lessons Lessons Popular Subjects algebra arithmetic calculus equations ... more References applied mathematics mathematical games mathematicians more ... Number theorists Helge von Koch 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 "Sur une courbe continue sans tangente, obtenue par une construction géométrique élémentaire". Reference The Plantagenet Roll of the Blood Royal (Mortimer-Percy Volume) by the Marquis of Ruvigny and Raineval (1911), pages 250 - 251

22. Snowflake Curve smaller and smaller triangles at each stage, is called the koch s SNOWFLAKECURVE, named after Niels Fabian helge von koch (Sweden, 18701924). http://scidiv.bcc.ctc.edu/Math/Snowflake.html

The Snowflake Curve 1. Start with an equilateral triangle whose sides have length 1. 2. On the middle third of each of the three sides, build an equilateral triangle with sides of length 1/3. Erase the base of each of the three new triangles. 3. On the middle third of each of the twelve sides, build an equilateral triangle with sides of length 1/9. Erase the base of each of the twelve new triangles. 4. Repeat the process with this 48-sided figure. See the likeness to a crystal of snow emerge? At the right, figure 4 is magnified by a power of two. The "limit curve" defined by repeating this process an infinite number of times, adding more and more, smaller and smaller triangles at each stage, is called the Koch's SNOWFLAKE CURVE , named after Niels Fabian Helge von Koch (Sweden, 1870-1924). The snowflake curve has some interesting properties that may seem paradoxical. The snowflake curve is connected in the sense that it does not have any breaks or gaps in it. But it's not smooth (jagged, even), because it has an infinite number of sharp corners in it that are packed together more closely than pebbles on a beach. n - 1 units are added at the nth step, so the length of the snowflake is larger than 3 + 1 + 1 + 1 + 1 + 1 + ....... = infinity.

23. Koch Curve -- Facts, Info, And Encyclopedia Article (Click link for more info and facts about helge von koch) helge von koch.The better known koch snowflake (or koch star) is the same as the curve, http://www.absoluteastronomy.com/encyclopedia/k/ko/koch_curve.htm

Koch curve [Categories: Fractals] Description The Koch curve is a (Click link for more info and facts about mathematical) mathematical (The trace of a point whose direction of motion changes) curve , and one of the earliest ((mathematics) a geometric pattern that is repeated at every scale and so cannot be represented by classical geometry) fractal curves to have been described. It appeared in a 1904 paper entitled "Sur une courbe continue sans tangente, obtenue par une construction géométrique élémentaire" by the (A Scandinavian language that is the official language of Sweden and one of two official languages of Finland) Swedish (A person skilled in mathematics) mathematician (Click link for more info and facts about Helge von Koch) Helge von Koch . The better known Koch snowflake (or Koch star ) is the same as the curve, except it starts with an (A three-sided regular polygon) equilateral triangle instead of a line segment. Eric Haines has developed the sphereflake fractal , a three- (Click link for more info and facts about dimensional) dimensional version of the (A crystal of snow) snowflake One can imagine that it was created by starting with a line segment, then recursively altering each line segment as follows:

24. Koch Snowflake -- From MathWorld A fractal, also known as the koch island, which was first described by helge vonkoch in 1904. Schneider, JE A Generalization of the von koch Curves. http://www.webmath.com/Answers/Files/problem_1504_0.htm

INDEX Algebra Applied Mathematics Calculus and Analysis Discrete Mathematics ... Alphabetical Index ABOUT THIS SITE About MathWorld About the Author Terms of Use DESTINATIONS What's New MathWorld Headline News Random Entry ... Live 3D Graphics CONTACT Email Comments Contribute! Sign the Guestbook MATHWORLD - IN PRINT Order book from Amazon Applied Mathematics Complex Systems Fractals Koch Snowflake 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 inner third of each side, building another equilateral triangle at the location where the side was removed, and then repeating the process indefinitely. The Koch snowflake can be simply encoded as a Lindenmayer system with initial string string rewriting rule and angle 60°. The zeroth through third iterations of the construction are shown above. The fractal can also be constructed using a base curve and motif, illustrated below.

25. What (Koch's Snowflake) The koch Curve was studied by helge von koch in 1904. When considered in itssnowflake form, (see below) the curve is infinitely long but surrounds finite http://www.shodor.org/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 is 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.

26. Lexikon Helge Von Koch helge von koch aus der freien http://lexikon.freenet.de/Helge_von_Koch

E-Mail Mitglieder Community Suche ... Hilfe document.write(''); im Web in freenet.de in Shopping Branchen Lexikon Artikel nach Themen alphabetischer Index Artikel in Kategorien Weitere Themen Doris gegen Angie: Neue Attacke im Krieg ums Kanzleramt Sex mit 50 Cent: Islands Frauen stehen Schlange Die Zukunft von Filesharing: Das kommt auf Sie zu Neuer Alfa im Test: Herrlich b¶se und aufregend maskulin ... "Mensch ¤rgere dich nicht:" Jetzt gratis mitspielen! Sie sind hier: Startseite Lexikon Helge von Koch Helge von Koch Niels Fabian Helge von Koch 25. Januar in Stockholm 11. M¤rz ebenda) war ein Schwedischer Mathematiker . Er konstruierte die nach ihm benannte Koch-Kurve , eines der ersten Fraktale , als Beispiel f¼r eine unendlich lange, an keiner Stelle differenzierbare Kurve. Helge von Koch wurde 1870 als Sohn des schwedischen Offiziers Richert Vogt von Koch und Agathe Henriette Wrede in Stockholm geboren. Nach der Schule studierte er an der Stockholmer Universit¤t , die damals noch H¶gskola (Hochschule) hieŸ, bei G¶sta Mittag-Leffler Mathematik. ver¶ffentlichte er eine Arbeit ¼ber die L¶sung von Differentialgleichungen , die zum Teil auf Vorarbeiten von Henri Poincare beruhte. Ein Jahr sp¤ter promovierte er mit einer Arbeit, die seine und Poincares Erkenntnisse umfasste. Von

27. Koch Doodles It was invented by a Swedish mathematician called helge von koch (18701924) andis usually called the koch snowflake. To see how to construct it, http://www.geocities.com/aladgyma/articles/scimaths/koch.htm

Koch Doodles One of the most famous fractals was invented long before the concept of a fractal was well-understood, or even understood at all. It was invented by a Swedish mathematician called Helge von Koch (1870-1924) and is usually called the Koch snowflake. To see how to construct it, take a line, divide it into thirds, and erect a triangle on the middle third. Next, take the four new lines and do the same to each of them. Et cetera ad infinitum. Play with these images to follow the process: Stage And if the line one begins with is one side of triangle, and the other two sides are treated in the same way, you get the Koch snowflake. Though it doesn’t end there, of course. One can use squares or rectangles instead of triangles, or both, and one can vary where one erects them and how high one erects them. The possibilities are endless, but you can get some flavor of them from the following: Image Stage Return to Subject Index Return to General Index Return to Maths Index

28. Biografia Di Niels Fabian Helge Von Koch - Caos E Oggetti Frattali - Eliana Arge Translate this page von koch, militare di carriera, e di Agathe Henriette Wrede, helge von koch von koch è famoso per la curva che porta il suo nome e che apparve nel 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

29. 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

30. Articles - Helge Von Koch Niels Fabian helge von koch (January 25, 1870 March 11, His father, RichertVogt von koch (1838-1913) was a Lieutenant-Colonel in the Royal Horse http://www.totalorange.com/articles/Helge_von_Koch

Educational Games Math Games Geography Games 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 "Sur une courbe continue sans tangente, obtenue par une construction g©om©trique ©l©mentaire". Reference The Plantagenet Roll of the Blood Royal (Mortimer-Percy Volume) by the Marquis of Ruvigny and Raineval (1911), pages 250 - 251 All text is available under the terms of the GNU Free Documentation License Source: Original text from the article in Wikipedia, The Free Encyclopedia:

31. NetLogo Models Library: Koch Curve helge von koch was a Swedish mathematician who, in 1904, introduced what is nowcalled the koch curve. Here is a simple geometric construction of the koch http://ccl.northwestern.edu/netlogo/models/KochCurve

Home Page Download Models Community Models ... Extensions User Manual: Web version Printable version FAQ Links ... Contact Us NetLogo Models Library: Sample Models/Mathematics/Fractals (back to the library) Koch Curve Run Koch Curve in your browser uses NetLogo 2.1 requires Java 1.4.1+ system requirements Note: If you download the NetLogo application, every model in the Models Library (besides the Community Models) is included. If you have trouble running this model in your browser, you may wish to download the application instead. WHAT IS IT? Helge von Koch was a Swedish mathematician who, in 1904, introduced what is now called the Koch curve. Here is a simple geometric construction of the Koch curve. Begin with a straight line. This initial object is also called the "initiator." Partition it into three equal parts. Then replace the middle third by an equilateral triangle and take away its base. This completes the basic construction step. A reduction of this figure, made of four parts, will be used in the following stages. It is called the "generator." Thus, we now repeat, taking each of the resulting line segments and partitioning them into three equal parts, and so on. The figure below illustrates this iterative process. Step 0: "Initiator"

32. Historia Matematica Mailing List Archive: Re: [HM] Koch Niels Fabian helge von koch Born 25 Jan 1870 in Stockholm, Sweden Died 11 March1924 in Stockholm, Sweden. helge koch was a student of MittagLeffler and http://sunsite.utk.edu/math_archives/.http/hypermail/historia/nov99/0119.html

Re: [HM] Koch Antreas P. Hatzipolakis xpolakis@otenet.gr Wed, 17 Nov 1999 02:04:22 +0200 (EET) Messages sorted by: [ date ] [ thread ] [ subject ] [ author ] Next message: James A. Landau: "Re: [HM] origins of logarithmic function" Previous message: Rene Grognard: "Re: [HM] Einstein's E=mc^2 was Italian's idea" Maybe in reply to: Wong Khoon Yoong: "[HM] Koch" Next in thread: Daniel E. Otero: "Re: [HM] Koch" Wong Khoon Yoong wrote: His short biography at St Andrews archive reads: Niels Fabian Helge von Koch Born: 25 Jan 1870 in Stockholm, Sweden Died: 11 March 1924 in Stockholm, Sweden Helge Koch was a student of Mittag-Leffler and succeeded him in 1911 at Stockholm University. He is famous for the Koch curve. This is constructed by dividing a line 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.

33. 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

34. 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.

35. 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.

36. 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

37. 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.

38. 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.

39. Koch's Snowflake, Mandelbrot's Coastline, Alaska Science Forum In 1904, the Swedish mathematician helge von koch described an interesting curiosity.He proposed a mental exercise that could be partially carried out in http://www.gi.alaska.edu/ScienceForum/ASF9/920.html

Alaska Science Forum April 5, 1989 Koch's Snowflake, Mandelbrot's Coastline Article #920 by Carla Helfferich This article is provided as a public service by the Geophysical Institute, University of Alaska Fairbanks, in cooperation with the UAF research community. Carla Helfferich is a science writer at the Institute. Journalists love numbers almost as much as scientists do, and probably for the same reason: if you can put an exact number on it, it must be real. That was never more clear than during the oil spill in Prince William Sound. Readers, listeners, and viewers were given numbers for everything from how many gallons are in a barrel of oil to the dollar value of the annual pink salmon catch. They were told how many sea otters lived near Naked Island and how many square miles of sea water lay under the oily scum. Yet in all the news of threatened coastline, rare indeed were statistics on the length of that coastline. At first, that doesn't seem logical. A coastline is obviously real, so it must be measurable in real numbers. Well, yes and no. Contemporary mathematics, with roots in the early part of this century, raises some doubts. In 1904, the Swedish mathematician Helge von Koch described an interesting curiosity. He proposed a mental exercise that could be partially carried out in visible form by anyone with pencil, paper, and patience.

40. 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

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