21. Koch Snowflake -- From MathWorld as the koch island, which was first described by helge von koch in 1904. The von koch Snowflake Curve Revisited. §C.2 in The Science of Fractal http://mathworld.wolfram.com/KochSnowflake.html

INDEX Algebra Applied Mathematics Calculus and Analysis Discrete Mathematics ... Alphabetical Index DESTINATIONS About MathWorld About the Author Headline News ... Random Entry CONTACT Contribute an Entry Send a Message to the Team 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 . 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. Let be the number of sides, be the length of a single side, be the length of the perimeter , and the snowflake's area after the th iteration. Further, denote the

22. Efg's Fractals And Chaos -- Von Koch Curve Lab Report Swedish mathematician helge von koch introduced the koch curve in 1904.Starting with a line segment, recursively replace the line segment as shown below http://www.efg2.com/Lab/FractalsAndChaos/vonKochCurve.htm

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

23. All About Helge Von Koch - RecipeLand.com Reference Library 1 A biography page of Niels Fabian helge von koch (http//wwwgap.dcs.st-and.ac.uk/~history/Mathematicians/koch.html) from the MacTutor History of http://www.recipeland.com/encyclopaedia/index.php/Helge_von_Koch

24. 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.

25. 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

26. 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.

27. 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.

28. 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

29. FRACTALES - "Matemática De Belleza Infinita" => INTRODUCCIÓN Al Concepto Fract Translate this page helge von koch no dejó a descubierto todos los misterios (o utilidades) de su isla.Antes de continuar, preferiría dejar como referencia su dimensión http://www.geocities.com/capecanaveral/cockpit/5889/koch2.html

LAS CURVAS DE KOCH: Lagos, Islas y otros En la búsqueda de nuevos fractales, puede hacerse una comparación (y al mismo tiempo complementar esta búsqueda) al analizar los distintos tipos de curvas de Koch existentes (como la ya conocida Isla Tríada de Koch). El estudio para la generación de estos cuerpos está enfocado a la medición de longitudes de diversos acervos naturales, si así puede llamárseles. Por eso, no debe parecer extraño encontrar definiciones que incluyan a lagos o cabos (y otros). La primera vez que vi estos conceptos me parecieron muy peculiares e incluso crei que se trataba de una broma o un juego de doble sentido (aún tengo mis dudas). Primero que nada, trateremos aspectos ya conocidos del Triángulo de Koch (o copo K o Isla Tríada, como guste) y profundizaremos más en su construcción, para poder sumergirnos en los mentados derivados de esta formación. Como sabrá, la Isla Tríada de Koch se genera a partir de un GENERADOR, que en este caso es un triángulo equilátero. Al colocar otro "invertido" sobre los tercios medios de sus lados, se forma la conocida "Estrella de David". Sobre cada uno de los seis triángulos nuevos se repite lo mismo, infinitas veces. El resultado es nuestro conocido copo K. Si bien es cierto que en la formación de este fractal existen puntos que nunca dejan de desplazarse, tarde o temprano llegan a un límite (o mucho mejor dicho, tienden a un límite) que termina por definir la costa que rodea nuestra isla.

30. 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:

31. Koch Tomorrow is Niels Fabian helge von koch s Birthday! koch. Born January 25in Stockholm, Sweden. Died March 11, 1924 near 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

32. First Day Of Winter - Koch The NCB thanks Niels Fabian helge von koch for his wonderful snowflakes. BornJanuary 25, 1870 in Stockholm, Sweden Died March 11, 1924 in Danderyd, http://curvebank.calstatela.edu/birthdayindex/dec/dec21koch/dec21koch.htm

Close Window Today officially marks the arrival of winter! The NCB thanks Niels Fabian Helge von Koch for his wonderful snowflakes. Born: January 25, 1870 in Stockholm, Sweden Died: March 11, 1924 in Danderyd, Stockholm, Sweden

33. 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"

34. Kochcurve Info Window 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/cm/models/kochcurve/info.html

35. Biografia Di Niels Fabian Helge Von Koch - Caos E Oggetti Frattali - Eliana Arge Translate this page helge von koch frequentò una buona scuola superiore di Stoccolma, von kochè famoso per la curva che porta il suo nome e che apparve nel suo lavoro 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

36. 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.

37. Koch Snowflake One of the most symmetric and easy to understand fractals; it is named after theSwedish mathematician helge von koch (18701924), who first described it in http://www.daviddarling.info/encyclopedia/K/Koch_snowflake.html

Return to The Worlds of David Darling INDEX LATEST NEWS ARCHIVE ... Z entire Web this site Koch snowflake One of the most symmetric and easy to understand fractals exterior snowflake , the Koch antisnowflake , and the flowsnake curves ALL MATHEMATICS ENTRIES BACK TO TOP var site="s13space1234"

38. 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.

39. 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 Britneys perverse Spende: Geburt im TV f¼r Hurrikan-Opfer Dirty Talk: K¼bl will Sex mit Ginny, Gina und Giuseppe Audi S4 vs. Porsche 4S: Zwei Sportler, zwei Charaktere Eiskalt verf¼hrt: Kuscheln im einzig- artigen Iglu-Hotel ... Download-Charts: Die beliebtesten Tools im œberblick 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.

40. The Koch Curve Above left we see the first four orders of the koch curve (drawn using Fractintand Paint Shop Pro), discovered by helge von koch. http://www.jimloy.com/fractals/koch.htm

Return to my Mathematics pages Go to my home page The Koch Curve Above left we see the first four orders of the Koch curve (drawn using Fractint and Paint Shop Pro ), discovered by Helge von Koch. Sometimes, a straight line segment is called the first order. And then the four images above left are the next four orders. You can probably see how each order is built from the previous one. Above right we see the third order Koch island (or snowflake), made up of three Koch curves. Below, is the fifth order Koch curve, magnified four times. The sixth order Koch curve (below) looks much like the fifth order, except that each tiny point is indistinct. It's hard to tell what is going on. Actually it is made up of many tinier points. But the resolution of the graphic image is inadequate to show points that small. The actual Koch curve (and island) is the limit of infinitely many orders. It looks like the picture below, again with inadequate resolution. You may have noticed that the Koch curve is very self-similar (see Fractals and Self-Similarity ). Various parts of it (the infinite order version) are identical to larger and smaller parts. So, each point that you see in the fifth order curve becomes a very convoluted portion of the curve in higher orders.

A  B  C  D  E  F  G  H  I  J  K  L  M  N  O  P  Q  R  S  T  U  V  W  X  Y  Z

Page 2     21-40 of 101    Back | 1  | 2  | 3  | 4  | 5  | 6  | Next 20