41. Igor Podlubny: Gallery: Sierpinski Carpet Igor Podlubny. Next Previous Exit sierpinski Carpet. waclaw sierpinski (1882–1969)created one of the most famous mathematical objects of the XXth century. http://www.tuke.sk/podlubny/sierpinski.html

This is what you would see in a standard-compliant browser: Igor Podlubny Next Previous Exit Sierpinski Carpet Waclaw Sierpinski fractal Using CSS only, I emulated the 4-th step of the creation of the Sierpinski carpet . There are no graphics elements in my emulation. You can check this by looking at the source code. The corresponding CSS code of my example is of particular interest: it has the property of self-similarity, exactly like the Sierpinski gasket itself. I reserve all rights for my examples of the power of CSS, and for the corresponding source code. You can study the source code for your education, but you are not allowed to distribute it in any form without my written permission. You can provide a link to my web page if you like it, but you are not allowed to place a copy at your web page. Valid XHTML 1.1 Valid CSS2 PLEASE NOTE: This site is made with XHTML 1.1 and CSS2. It will look much better in a browser that supports current web standards , but it is still accessible to any browser or Internet device.

42. The Science Bookstore - Books sierpinski, waclaw. Lecons sur les Nombres Transfinis. Paris GauthierVillars,1928. 1st edition. sierpinski, waclaw. Lecons sur les Nombres Transfinis. http://www.thesciencebookstore.com/bookmain.asp?pg=5&bookcat=Mathematics

43. Weekly List Of Books xvi, 731p. ISBN 007-044758-6. 511.8 P01;1 G16772 ** Mathematical modelling27 sierpinski, waclaw General topology by waclaw sierpinski. http://www.library.iisc.ernet.in/access/wklstbks/11nov2k2.html

J R D TATA MEMORIAL LIBRARY WEEKLY LIST OF BOOK ADDITIONS From To Part I : Main Library Part II: Department Libraries Part I: Main Library Serial No: Book catalogue ASTRONOMY 1 Yang, Qihe H. Map projection transformation: Principles and applications: by Qihe H. Yang, John P. Snyder and Waldo R. Tobler. London: Taylor and Francis, 2000. xv, 367p. ISBN : 0-748-40668-9. 526.8 P R(T) 169993 ** Map projections CHEMISTRY AND ALLIED SCIENCES 2 Gutsche, David C. Calixarenes revisted: by C. David Gutsche. Cambridge: Royal Society of Chemistry, 1998. xii, 233p. (Monographs in Supramolecular Chemistry; no. 1). ISBN : 0-85404-502-3. ** Calixarenes; Phenols; Macromolecular compounds COMPUTER SCIENCE 3 Buford, John F. Koegel, ed. Multimedia systems: ed by John F. Koegel Buford. New York: ACM Press, 2000. xii, 450p. ISBN : 81-7808-162-8. 006 P001 R(T) 169955 ** Multimedia systems; Multimedia systems-architecture 4 Worboys, Michael, F. GIS: A computing perspective: by Michael F. Worboys. London: Taylor and Francis, 1995. xiv, 376p. ISBN : 0-7484-0065-6.

44. Triângulo De Sierpinski Translate this page figura seguinte. Este triângulo foi descrito por waclaw Sierpinskiem 1915 e obtem-se como limite de um processo recursivo. Para http://www.educ.fc.ul.pt/icm/icm99/icm48/sierpinski.htm

Triângulo de Sierpinski O Triângulo de Sierpinski pertence a uma classe de objectos matemáticos conhecidos como fractais, cuja principal característica é não perder a sua definição inicial à medida que é ampliado. Esta característica é bem visível na figura seguinte. Este triângulo foi descrito por Waclaw Sierpinski em 1915 e obtem-se como limite de um processo recursivo. Para começar o processo partimos de um triângulo equilátero. Em seguida unem-se os pontos médios de cada lado do triângulo, formando 4 triângulos cujos lados estão ligados. Retira-se agora o triângulo central. A recursão consiste em repetir indefenidamente o procedimento anterior em relação a cada um dos triângulos obtidos. Qual a relação com o triângulo de Pascal? Se retirarmos os números pares e colorirmos de preto os números impares obtemos a seguinte imagem. O triângulo de Pascal "transforma-se" assim no triângulo de Sierpinski. Mas esta não é a única configuração interessante que podemos obter com o triângulo de Pascal.

45. Sierpinski Problem sierpinski proved there exist infinitely many odd integers k such that k*2^n+1 is composite for every Category Science Math Number Theory Factoring......The sierpinski Problem Definition and Status. In 1960 waclaw sierpinski(18821969) proved the following interesting result. Theorem http://www.prothsearch.net/sierp.html

The Sierpinski Problem: Definition and Status In 1960 Waclaw Sierpinski (1882-1969) proved the following interesting result. Theorem [S] There exist infinitely many odd integers k such that k n + 1 is composite for every n A multiplier k with this property is called a Sierpinski number . The Sierpinski problem consists in determining the smallest Sierpinski number. In 1962, John Selfridge discovered the Sierpinski number k = 78557, which is now believed to be in fact the smallest such number. Conjecture. The integer k is the smallest Sierpinski number. To prove the conjecture, it would be sufficient to exhibit a prime k n + 1 for each k Summary of results. This summary describes developments in the computational approach to a possible "solution" of the Sierpinski problem, from the earliest attempts in the late 1970ies until November 2002, and gives a comprehensive status of results known at that point. For more recent information, refer to the distributed computing project Seventeen or Bust . The name of this project indicates that when it was created, only 17 uncertain candidates k were left to be investigated, namely

46. Sierpinski Pyramid waclaw sierpinski (18821969) was a professor at Lvov and Warsaw.He was one of the most influential mathematicians of his time http://www.bearcave.com/dxf/sier.htm

Sierpinski Pyramid This Web page publishes the C++ code that generates a 3-D object that I call a Sierpinski pyramid. The Sierpinski pyramid program displays a wire frame of the pyramid, and rotates it through all three dimensions, using openGL. A DXF description for the object is written to a file or to stdout . The DXF file format was developed by AutoDesk and is commonly used to exchange 3-D models. Most 3-D rendering programs can read DXF format files. The Sierpinski pyramid is inspired by the two dimensional Sierpinski "gasket" described in Chaos and Fractals: New Frontiers of Science by Peitgen, Jurgens and Saupe, Springer Verlag 1992. Waclaw Sierpinski (1882-1969) was a professor at Lvov and Warsaw. He was one of the most influential mathematicians of his time in Poland and had a worldwide reputation. In fact, one of the moon's craters is named after him. The basic geometric construction of the Sierpinski gasket goes as follows. We begin with a triangle in the plane and then apply a repetitive scheme of operations to it (when we say triangle here, we mean a blackened, 'filled-in' triangle). Pick the midpoints of its three sides. Together with the old verticies of the original triangle, these midpoints define four congruent triangles of which we drop the center one. This completes the basic construction step. In other words, after the first step we have three congruent triangles whose sides have exactly half the size of the original triangle and which touch at three points which are common verticies of two contiguous trianges. Now we follow the same procedure with the three remaining triangles and repeat the basic step as often as desired. That is, we start with one triangle and then produce 3, 9, 27, 81, 243, ... triangles, each of which is an exact scaled down version of the triangles in the preceeding step.

47. Sierpinski Triangle waclaw sierpinski (1882 1969 ) World War I totally disrupted themathematical communities of eastern Europe. Rather than try to http://curvebank.calstatela.edu/sierpinski/sierpinski.htm

Back to . . . Curvebank Home Page Sierpinski Triangles This requires JAVA 1.2 or better. If you have a Mac, the operating system must be OS X or newer. Using the mouse, click on any three points in the box. NCB Deposit # 2 The National Curve Bank welcomes the Sierpinski Triangle animation of Kathleen Shannon and Michael Bardzell Dept. of Mathematics and Computer Science Salisbury University, Salisbury, MD. NCB Deposit #3 A Sample of recursion equations: For a definition click here: Click on the stamp to see an enlargement. Click on the Mandelbrot Set to read more about fractals. Waclaw Sierpinski World War I totally disrupted the mathematical communities of eastern Europe. Rather than try to re-build comprehensive university programs in several areas of research, Sierpinski, Kuratowski, Banach and others decided to work together in the emerging field of abstract spaces. They soon became known as the "Polish School." Their first international recognition came from publishing a new journal, Fundamenta Mathematicae (1920), devoted to set theory and related topics, and not to their work in topology. Indeed, the publication of Banach's dissertation in 1922 has been called the birth of functional analysis.

48. The Sierpinski Gasket On the left and right, we see a series of sierpinski gaskets (drawn usingFractint and Paint Shop Pro), discovered by waclaw sierpinski. http://www.jimloy.com/fractals/sierpins.htm

Return to my Mathematics pages Go to my home page The Sierpinski Gasket A Sierpinski gasket is also called a Sierpinski sieve. On the left and right, we see a series of Sierpinski gaskets (drawn using Fractint and Paint Shop Pro ), discovered by Waclaw Sierpinski. The first order would just be a straight line segment. Here, I show orders 2 through 7. You should probably see how each new order is built from the previous one. The true Sierpinski gasket is the limit of infinitely many of these steps. Instead of lines, we can also build it with dark triangles (or any other object). The Sierpinski gasket is related to The Yanghui Triangle (usually called Pascal's triangle), below. I have drawn hexagons around the odd numbers. That pattern is identical to that of the Sierpinski gasket, forever. There is an interesting experiment called "the chaos game," in which random (presumably chaotic) chance produces great order. On the left, we see a picture. We draw three (or more) points (the vertices of a triangle, which doesn't have to be equilateral or isosceles), labeled 1, 2, and 3. Then we choose a starting point S, at random (the one I chose is not within the triangle). Then we begin the game. We proceed to choose random numbers, 1, 2, or 3 (with dice or whatever). Each random number defines a new point halfway between the latest point and the point toward which our random number directs us. For example, my first random number was a 1; so I drew a point halfway between S and 1. Then I got another random 1, then 3, 2, 1, and 3. After drawing 6 points, I perceive no obvious pattern. With a computer, it is easier to continue to choose many more points.

49. Efg's Fractals And Chaos -- Sierpinski Triangle Lab Report Mathematical Background. Polish mathematician waclaw sierpinski introducedthe sierpinski Gasket in 1915. Starting with a triangle http://homepages.borland.com/efg2lab/FractalsAndChaos/SierpinskiTriangle.htm

Fractals and Chaos Sierpinski Triangle Lab Report Create a Sierpinkski "Gasket" By Cutting Holes in a Triangle Purpose The purpose of this project is to show how to create a Sierpinksi gasket, a "holey" triangle, by recursively cutting holes in a triangle. Mathematical Background Polish mathematician Waclaw Sierpinski introduced the "Sierpinski Gasket" in 1915. Starting with a triangle, recursively cut the triangle formed by the midpoints of each side: The single equilateral triangle in Step 0, is divided into four equal-area equilateral triangles in Step 2. The "middle" triangle is colored differently to indicate it has been "cut" from the object. This same "rule" is applied an infinite number of times. Here are the next two steps: Let's analyze what's happening. Consider the perimeter of the red triangles: Step Triangles 3 sides/triangle Length of Side Total Length a a/2 a/4 k k a/2 k k+1 a/2 k As k approaches infinity, the perimeter of all the red triangles approaches infinity. The area of an equilateral triangle with each side length a is . (See the von Koch Curve Lab Report for details.) For further computations here, we'll make computations as a fraction of A

50. The Wu! © Fractals: The Sierpinski Triangle Polish mathematician waclaw sierpinski (18821969) worked in the areas of set theory,topology and number theory, and made important contributions to the axiom http://www.ocf.berkeley.edu/~wwu/fractals/sierpinski.html

Fractals: The Sierpinski Triangle Home Introduction to Fractals Mandelbrot Sierpinski ... Gallery The Sierpinski Triangle is the orbit S of a seed in the Chaos Game. Polish mathematician Waclaw Sierpinski (1882-1969) worked in the areas of set theory, topology and number theory, and made important contributions to the axiom of choice and continuum hypothesis. But he is best known for the fractal that bears his name, the Sierpinski triangle, which he introduced in 1916. The Sierpinski triangle, sometimes referred to as the Sierpinski gasket, is a simple iterated function system that often serves as the first example of a fractal given to elementary school or high school students. There are two main ways to construct the triangle, one of which is obvious, and the other rather incredible. Construction 1 : Begin with a base triangle, and then draw lines connecting the midpoints of each leg, forming three self-similar right-side up subtriangles at each of the base triangle's corners. Then repeat this process for each of the newly formed subtriangles, and so on, ad infinitum. Construction 2 : "The Chaos Game" Choose three random points A, B, and C in some plane P, and color one of them red, another blue, and the third green. We will refer to these points as vertices, since one can imagine them as vertices of a triangle.

51. área Fractal - Koch & Sierpinski Translate this page Alrededor de 1915, waclaw sierpinski (1862-1969) concibió su archiconocidofractal (fig. 2). Partiendo de un triángulo (no tiene 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

52. Untitled (Bunde). waclaw sierpinski was born on March 14, 1882, in Warsaw, Poland. waclawsierpinski (The Mactutor). he would spend the rest of his life. http://www.facstaff.bucknell.edu/udaepp/090/w3/toddw.htm

Todd Wenrich Prof. Daepp Final exam Fractal Geometry Waclaw Sierpinski was born on March 14, 1882, in Warsaw, Poland. Sierpinski attended the University of Warsaw in 1899, when all classes were taught in Russian. He graduated in 1904 and went on to teach mathematics and physics at a girl's school in Warsaw. He left teaching in 1905 to get his doctorate at the Jagiellonian University in Cracow. After receiving his doctorate in 1908, Sierpinski went on to teach at the University of Lvov. During his years at Lvov, he wrote three books and many research papers. These books were The Theory of Irrational numbers Outline of Set Theory (1912), and The Theory of Numbers (1912). In 1919, Sierpinski accepted a job as a professor at the University of Warsaw, and this is where Waclaw Sierpinski (The Mactutor) he would spend the rest of his life. Throughout his career, Sierpinski wrote 724 papers and an amazing 50 books. Sierpinski studied many areas of mathematics, including, irrational numbers, set theory, fractal geometry, and theory of numbers. Sierpinski is viewed as one of the greatest Polish mathematicians ever. He is noted for his construction of the Sierpinski gasket. (The Mactutor) Sierpinki gasket (The MacTutor) Sierpinski Carpet (Bunde) The Sierpinski carpet for the above example has n = 5 and k = 9. It is possible for the carpets to look different when k is changed. Using the same idea as for the calculation of the dimension of the gasket, we denote by L the length of a square side and by the M mass of the carpet. Considering that n^2 – k smaller squares with side length L/n make up the whole carpet with side length L we get M(L) = (n^2 – k) M (L/n). Combining this with the general formula M = A L^d for some constant A we get (n^2 – k) A (L/n)^d = A L^d which simplifies to n^2 – k = n^d. Taking logarithms on both sides and solving for the dimension we get d = log(n^2 –k) / log n. In the example above, the fractal dimension is log(16) / log(5) = 1.7227 .(Daepp)

53. Dolls waclaw sierpinski (18821969) was a prolific Polish mathematician specialized innumber theory, who created and studied several self-similar patterns and the http://odur.let.rug.nl/~koster/dolls.htm

Self-similar structures The intriguing pattern on this picture is a floor mosaic found in the cathedral of Anagni (Italy). The cathedral and the floor were constructed in the year 1104 (information provided by Nicoletta Sala It is a spectacular early example of what is known in mathematics as an iterative function system, a function that can be recursively iterated (repeated) to create fractal -like structures. The most important property of these functions is that they create self-similar structures, i.e., structures with parts that have the same form as the entire structure. On the picture, you see that the biggest triangles contain the next biggest triangles, surrounded by three triangles of the same kind, containing the next biggest triangle, etc. This process can be infinitely repeated. If you find such structures and ideas complicated, just consider the following picture, which makes it all clear (with thanks to BU's math department): This picture shows how the triangle patterns, so-called Sierpinski triangles, are generated. There are even web sites with Java applets, which allow you to interactively generate these patterns Waclaw Sierpinski (1882-1969) was a prolific Polish mathematician specialized in number theory, who created and studied several self-similar patterns and the functions generating them. The triangles named after him are the most famous example. Fascinating as they are, Sierpinski triangles received much attention during the last few decades, thanks to the world-wide attention for

54. GalaxyGoo Math Links: Mathematicians, Historical Plato; Ptolemy, Claudius; Pythagoras of Samos; Riemann, Georg FriedrichBernhard; sierpinski, waclaw; Taylor, Brook; Turing, Alan Mathison; http://www.galaxygoo.org/blogs/archives_math/000153.html

GalaxyGoo Math Links Links to Math Resources Main December 14, 2002 Mathematicians, Historical Biographies of Women Mathematicians Indexes of Biographies Archimedes of Syracuse Aristotle ... Weierstrass, Karl Theodor Wilhelm Posted by Richard at December 14, 2002 04:10 PM Comments Post a comment Name: Email Address: URL: Comments: Remember info?

55. Sierpinski This figure, called the sierpinski triangle, is named after the Polish mathematicianwaclaw sierpinski, who described the object in 1917. waclaw sierpinski. http://www.templejc.edu/precalc/media/units/unit1/Topic2/RefGuide2/Files/History

Historical Note This figure, called the Sierpinski triangle, is named after the Polish mathematician Waclaw Sierpinski, who described the object in 1917. Waclaw Sierpinski [More information to come] Return to the Reference Guide.

56. Untitled waclaw sierpinski was a Polish mathematician of the XX century very known inhis country but also all around the world in the mathematics history and http://www.enseeiht.fr/hmf/travaux/CD9900/travaux/optmfn/hi/00pa/mfn03/biograp.h

57. What (Sierpinski's Triangle) Turn of the century mathematician waclaw sierpinski's name was given to severalfractal objects, the most famous being his Triangle or Gasket. http://www.ecu.edu/si/cd/interactivate/activities/gasket/what.html

What is the Sierpinski's Triangle Activity? This activity allows the user to step through the process of building the Sierpinski's Triangle. Turn of the century mathematician Waclaw Sierpinski's name was given to several fractal objects, the most famous being his Triangle or Gasket. This surface is idiosyncratic in that it has no area. To build the Sierpinski's Gasket, start with an equilateral triangle with side length 1 unit, completely shaded. (Iteration 0, or the initiator) Cut out of each triangle the smaller triangle formed by connecting the midpoints of each of the sides. (the generator) Repeat this process on all shaded triangles. Stages 0, 1 and 2 are shown below. The limiting figure for this process is called the Sierpinski's Gasket. It is one of the classic regular fractals Class Resources Exploration Questions Discussion on Plane Figure Fractals Sierpinski's Carpet Activity Please direct questions and comments about this page to Interactivate@shodor.org

58. Courbe De Sierpinski Translate this page COURBE DE sierpinski sierpinski's curve, sierpinskische Kurve. Courbe étudiée parsierpinski en 1912. waclaw sierpinski (1882-1969) mathématicien polonais. http://www.mathcurve.com/fractals/sierpinski/sierpinskicourbe.shtml

fractal suivant courbes 2D courbes 3D surfaces ... fractals COURBE DE SIERPINSKI Sierpinski's curve, Sierpinskische Kurve Waclaw Sierpinski La courbe de Sierpinski est une courbe remplissant Peano Hilbert comme le montre les figures suivantes : Ne pas confondre cette courbe avec la courbe du triangle de Sierpinski Figure kolam traditionnelle indienne fractal suivant courbes 2D courbes 3D ... Jacques MANDONNET

59. Triangle De Sierpinski sierpinski gasket (soit joint de culasse de sierpinski ), estdue à Mandelbrot. waclaw sierpinski (1882-1969) mathématicien polonais. http://www.mathcurve.com/fractals/sierpinski/sierpinskitriangle.shtml

fractal suivant courbes 2D courbes 3D surfaces ... fractals TRIANGLE DE SIERPINSKI Sierpinski gasket, Sierpinski-Gasket Autre nom : tamis de Sierpinski. Waclaw Sierpinski Le triangle de Sierpinski est un fractal de Sierpinski C'est l' attracteur Le triangle de Sierpinski est aussi la limite d'une suite de courbes continues sans point double dites courbes du triangle de Sierpinski (en anglais, arrowhead curves) : Les coefficients impairs sont sur les cases rouges et les pairs sur les blanches ! Sierpinski coke Coquillage de Sierpinski fractal suivant courbes 2D courbes 3D surfaces ... Jacques MANDONNET

60. LogBank - On Logic In Warsaw Scientific Society played a significant role for mathematics owing to prompt publication of results;their first volume, including some waclaw sierpinski's results, appeared in http://www.calculemus.org/LogBank/HTPL/SocScVars.html

Log Bank On the Warsaw Scientific Society and Its Contributions to Logic ARSAW SCIENTIFIC SOCIETY, whose official name in Latin reads Societas Scientiarum Varsaviensis , was established in Warsaw, 1800, to advance sciences and arts. It is merited for enormous contributions to scientific and cultural development of Poland. For instance, in the twenties of the 19th century the mechanical calculator of Abraham Stern (a Jewish mechanician from a small town in Eastern Poland) was demonstrated by the constructor and discussed at its sessions. In the first half of our century, its merits are most conspicuous in logic and mathematics. The Proceedings of its sessions played a significant role for mathematics owing to prompt publication of results; their first volume, including some Waclaw Sierpinski's results, appeared in 1908. Consequential achievements, e.g., those on definability, were first announced at its sessions, and seminal works were first published by it, as Jacques Herbrand's Recherches sur la Theorie de la Demonstration , 1930, and Alfred Tarski's Pojecie Prawdy (The Concept of Truth in Formalized Languages), 1933.

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