61. [ Wu :: Fractals | Sierpinski ] Polish mathematician waclaw sierpinski (18821969) worked in the areas of settheory, topology and number theory, and made important contributions to the http://www.ocf.berkeley.edu/~wwu/fractals/sierpinski.html

Sierpinski Triangle how to generate the sierpinski triangle fractal both deterministically and via the randomized chaos game. Fractal Intro Mandelbrot Sierpinski Gallery ... Return to Homepage 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.

62. ScienceNow In 1960, Polish mathematician waclaw sierpinski discovered, surprisingly, thatthe expression k * 2n + 1 was never prime when certain values of k were used, http://bric.postech.ac.kr/science/97now/02_12now/021219b.html

19 December 2002 Prime Riddle "Seventeen or bust" was the rallying cry. Now it's "thirteen or bust." A rag-tag group of math aficionados have had dramatic success in the past week and are well on their way to cracking a well-known conjecture in number theory: the Sierpinski Problem. In 1960, Polish mathematician Waclaw Sierpinski discovered, surprisingly, that the expression k * 2 n + 1 was never prime when certain values of k were used, no matter what natural number n was plugged into the formula. These values of k are known as Sierpinski numbers. "There's no obvious reason why they should exist," says Chris Caldwell, a mathematician at the University of Tennessee, Martin. "For example, if you look at 3 * 2 n + 1, it dumps out primes quite regularly, and the obvious feeling was they all should." But Sierpinski proved that the number 78,557and an infinite number of others now known as Sierpinski numbersalways spit out nonprimes. Furthermore, the structure of Sierpinski's proof implied that 78,557 was the smallest such number, but nobody really knew for sure whether this was the case. This is the Sierpinski Problem: Is 78,557 truly the smallest Sierpinski number? Mathematicians knew of 17 numbers smaller than 78,557 that might be Sierpinski numbers. To narrow down the list, a team of about 1000 volunteers recently dedicated the unused processing power of their computers. Led by computer science student Louis Helm of the University of Michigan, Ann Arbor, and programmer David Norris of the University of Illinois, Urbana-Champaign, the "Seventeen or Bust" collaboration began systematically testing each of the candidates. Between 27 November and 10 December, the team had proved that four of the candidates generated prime numbersproving that they are not Sierpinski numbers. "It was a very exciting week," says Helm. Getting four primes in rapid succession was "basically luck," says Norris, who expects that knocking down the 13 remaining Sierpinski candidates will take the better part of a decade.

63. NewPoland - Famous Poles: Scientists sierpinski, waclaw (18821969), mathematician. He was a father of the famousPolish School of Mathematics. His most important works are in the area of set http://www.newpoland.com/famous_poles_scientists_main.htm

NewPoland Mail Free email for life NewPoland News Stay informed! Home About Us Contact Us Testimonials ... Photo Gallery What's Related? Architecture Kings Major Dates Polish Cities ... Genealogy Hello! May I help you? Tell a Friend! Spread the word about NewPoland. Tell a Friend! Add a Site! Want to add a URL? It's simple. Add here! Connect with Us! Link to us Affiliates Become an affiliate and get more traffic. Partners We are currently seeking partners in various areas. Home Famous Poles Polish Contributors to World Sciences Famous Poles Polish Contributors to the Development of World's Sciences Banach, Stefan

64. Math Forum - Ask Dr. Math What is sierpinski s triangle? Date 01/20/97 at 112636 From Doctor TobySubject Re sierpinski Triangle waclaw sierpinski invented the triangle (or http://mathforum.org/library/drmath/view/54524.html

Associated Topics Dr. Math Home Search Dr. Math Sierpinski Triangle Date: 01/15/97 at 10:28:11 From: Anonymous Subject: Sierpinski Triangle Hi, My name is Ryan and I would like to ask you a question. What is a Sierpinski Triangle? http://mathforum.org/dr.math/ Associated Topics High School Fractals Search the Dr. Math Library: Find items containing (put spaces between keywords): Click only once for faster results: [ Choose "whole words" when searching for a word like age. all keywords, in any order at least one, that exact phrase parts of words whole words Submit your own question to Dr. Math Math Forum Home Math Library Quick Reference ... Math Forum Search Ask Dr. Math TM http://mathforum.org/dr.math/

65. Science Jokes:Waclaw Sierpinsky waclaw Sierpinsky. waclaw Sierpinsky (18821969), Polish mathematician.counting trunks Index Comments and Contributions. http://www.xs4all.nl/~jcdverha/scijokes/Sierpinski.html

Index Comments and Contributions Index Jokes with Famous Scientists Waclaw Sierpinsky Waclaw Sierpinsky (1882-1969), Polish mathematician counting trunks Index Comments and Contributions

66. Wacław Sierpiński - Encyklopedia waclaw sierpinski. Z Wikipedii, wolnej encyklopedii. waclaw Franciszek sierpinski (ur.14 marca 1882 r. w Warszawie zm. 21 pazdziernika 1969 r. w http://encyklopedia.korba.pl/wiki/Wacław_Sierpiński

Startuj z Nami! Dodaj do ulubionych Korba.pl Polski internet ... Encyklopedia Szukaj w encyklopedi: Kategorie stron Polscy matematycy Wacław Sierpiński Z Wikipedii, wolnej encyklopedii. Wacław Franciszek Sierpiński (ur. 14 marca r. w Warszawie - zm. 21 października r. w Warszawie) - polski matematyk Urodził się w rodzinie Konstantego, lekarza, i Ludwiki z Łapińskich. W ukończył V Gimnazjum Klasyczne w Warszawie i w tym też roku rozpoczÄ Å‚ studia na Wydziale Fizyko-Matematycznym Cesarskiego Uniwersytetu Warszawskiego. W zakończył studia, uzyskujÄ c stopień kandydata nauk i złoty medal za pracę z teorii liczb na temat podany przez prof. G. F. Woronoja, a od jesieni został mianowany nauczycielem matematyki i fizyki w IV Gimnazjum Å»eńskim. Uczestniczył w strajku szkolnym w , porzucił pracę i wyjechał do Krakowa , gdzie kontynuował studia na Wydziale Filozoficznym Uniwersytetu Jagiellońskiego W uzyskał stopień doktora filozofii. Po powrocie do Warszawy uczył w polskich szkołach średnich prywatnych, w Seminarium Nauczycielskim w Ursynowie oraz wykładał matematykę na Wyższych Kursach Naukowych, będÄ cych odpowiednikiem nieoficjalnego Uniwersytetu Polskiego w Warszawie. W wyjechał na kilkumiesięczne studia do Getyngi, gdzie zetknÄ Å‚ się z C. Caratheodorym. W styczniu 1908 został członkiem Towarzystwa Naukowego Warszawskiego, a w lipcu habilitował się na

67. Sierpinski Pyramid waclaw sierpinski (18821969) was a professor at Lvov and Warsaw. He was one ofthe most influential mathematicians of his time in Poland and had a 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.

68. NEW MATH BOOKS—UTICA CAMPUS LIBRARY Seiter, Charles. Everyday Math for Dummies. (QA 36 .S45 1995). sierpinski, waclaw.Pythagorean Triangles. (QA 460 .P8 S5313 2003). Smeltzer, Donald. http://www.mvcc.edu/library/acquisitions_math.html

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69. Sierpinski Triangle -- Facts, Info, And Encyclopedia Article and so cannot be represented by classical geometry) fractal, named after (Clicklink for more info and facts about waclaw sierpinski) waclaw sierpinski. http://www.absoluteastronomy.com/encyclopedia/s/si/sierpinski_triangle.htm

Sierpinski triangle [Categories: Fractals] The Sierpinski triangle , also called the Sierpinski gasket , is a ((mathematics) a geometric pattern that is repeated at every scale and so cannot be represented by classical geometry) fractal , named after (Click link for more info and facts about Waclaw Sierpinski) Waclaw Sierpinski An algorithm for obtaining arbitrarily close approximations to the Sierpinski triangle is as follows: 1. Start with any triangle in a plane. The canonical Sierpinski triangle uses an (A three-sided regular polygon) equilateral triangle with a base parallel to the horizontal axis, (first image). 2. Shrink the triangle by 1/2, make three copies, and position the three copies so that each triangle touches the two other triangles at a corner, (image 2). 3. Repeat step 2 with each of the smaller triangles, (image 3 and so on). Note that this infinite process is not dependent upon the starting shape being a triangle - it is just clearer that way. The first few steps starting, for example, from a square also tend towards a Sierpinsky gasket. (Click link for more info and facts about Michael Barnsley) Michael Barnsley was using an image of a fish to illustrate this in his paper PrintLink("http://wwwmaths.anu.edu.au/~barnsley/pdfs/V-var_super_fractals.pdf", "V-variable fractals and superfractals")

70. Sierpinski Number -- Facts, Info, And Encyclopedia Article In 1960 (Click link for more info and facts about waclaw sierpinski) waclawsierpinski proved that there are (Click link for more info and facts about http://www.absoluteastronomy.com/encyclopedia/S/Si/Sierpinski_number.htm

Sierpinski number [Categories: Analytic number theory] In (A science (or group of related sciences) dealing with the logic of quantity and shape and arrangement) mathematics , a Sierpinski number is an odd (The number 1 and any other number obtained by adding 1 to it repeatedly) natural number k such that integers of the form k n + 1 are composite (i.e. not (A number that has no factor but itself and 1) prime ) for all natural numbers n In other words, when k is a Sierpinski number, all members of the following set are composite: In 1960 (Click link for more info and facts about Waclaw Sierpinski) Waclaw Sierpinski proved that there are (Click link for more info and facts about infinite) infinite ly many odd (Any of the natural numbers (positive or negative) or zero) integer s that when used as k produce no primes. The Sierpinski problem is: "What is the smallest Sierpinski number?" In 1962, (Click link for more info and facts about John Selfridge) John Selfridge proposed what is known as Selfridge's conjecture : that 78,557 was the answer to the Sierpinski problem. Selfridge found that when 78,557 was used as

71. Sierpinski Curve@Everything2.com The sierpinski curve, invented by waclaw sierpinski, is a planefilling,non-intersecting fractal. It has the rather odd properties of being infinitely long http://www.everything2.com/index.pl?node_id=836306

72. ¿ÍÅ©·Î¿ì ½Ã¾îÇɽºÅ°(Waclaw Sierpinski) Translate this page The summary for this Korean page contains characters that cannot be correctly displayed in this language/character set. http://user.chollian.net/~badang25/sierpinski/waclaw_sier.htm

¹Ùú¶óÇÁ ½¿¡¸£ÇɽºÅ°(Waclaw Sierpinski) Warsaw, Æú¶õµå ½¿¡¸£ÇɽºÅ°ÀÇ °¡Àå Áß¿äÇÑ ¾÷ÀûÀº area of set theory(ÁýÇÕ·ÐÀÇ ¿µ¿ª), point set topolosy, number theory(Á¤¼ö·Ð)¿¡¼­ ÀÌ´Ù. ÁýÇշп¡¼­ ±×´Â the axiom of choice(¼±ÅÀÇ °ø¸®)¿Í the continuem hypothesis(¿¬¼Ó¼ °¡¼³)¿¡ °øÇåÇÏ¿´´Ù. Waclaw Sierpinski ´Â "¹Ùú¶óÇÁ ½¿¡¸£ÇɽºÅ°"¶ó°í Àд´ٰí ÇÕ´Ï´Ù. Warsaw´Â Æú¶õµåÀÇ ¼öµµ "¹Ù¸£»þ¹Ù"ÀÔ´Ï´Ù.

73. In The Beginning There Were Algebras Of Concrete Relations sierpinski, waclaw. Smith, Edgar C., Jr. Szczerba, Leslaw W. Szmielew, Wanda.Vaught, Robert L. A real quote (according to MacTutor) of Tarski http://www1.chapman.edu/~jipsen/talks/Tarski2001/Tarskitalk.htm

The many descendants of Tarski’s Relation Algebras Peter Jipsen Vanderbilt University Alfred Tarski Centenary Conference, Warsaw, May 29, 2001 A story about the creation of Relation Algebras In the beginning there were algebras of concrete relations. Tarski saw they were good, and he separated the interesting ideas from the trivial ones. And Tarski said “Let there be an abstract theory about these algebras”. So he made the theory of Relation Algebras. And he saw it was good. And then Tarski said “Let the theory produce all the known results about concrete relations”. And it was so. And he proved many interesting new results about relation algebras, including a correspondence with 3-variable logic that allowed the interpretation of set theory and he provided the first example of an undecidable equational theory. And Tarski said “Let the minds teem with new conjectures, let ideas fly, and let the community produce many new related theories and results”. Thus the field of relation algebras was born, with its many applications and connections to other areas. (all quotes fictitious; passage based on well known source)

74. VEDA MATEMATIKOVÉ V HISTORII waclaw sierpinski Jirí Svršek. narozen 14. brezna 1882ve Varšave, Polsko zemrel 21. ríjna 1969 ve Varšave, Polsko http://pes.internet.cz/veda/clanky/16529_48_0_0.html

NEVIDITELNÝ PES ZVÍØETNÍK VÌDA SCI-FI ... ENCYKLOPEDIE Nedìle 11.11.2001 Svátek má Martin Biologie a pøíroda Vesmír Fyzika Medicína ... Archiv vydání Nadpis Autor Text èlánku Mìjte rádi pejsky NEJKRÁSNÌJŠÍ VÁNOÈNÍ DÁREK PØESTAÒ KOUØIT A VYHRAJEŠ VLASTNÍ ŽIVOT akce zaèíná na serveru HTTP://viditelne.prase.cz Motto akce: "Politik nemùže pøijímat do státního rozpoètu peníze získané prodejem nejnebezeènìjší drogy cigarety, která zabije jen v Èeské Republice dvacet tisíc lidí roènì, absurdnì v dobì, kdy policie a záchranné sbory musí a po právu prohlížet kupøíkladu obálky, ale i všeliké lokality, které jsou by jen podezøelé z toho, že by mohly být kontaminovány nebezpeènou nákazou. Policie nás chrání pøed terorem a patøí jí za to dík i ohleduplnost nás všech." Smrt a nemoci z cigaret nejsou legální tím, že je brání zkorumpovaní politikové, leckdy bohužel i prostou ignorací žurnalisté, èi lobisté tabákových koncernù. Smrt z cigaret je stejný teroristický èin jako kterýkoliv jiný a po duchu platných zákonù je veøejným ohrožením èíslo jedna. Což ví ostatnì i pan ministr zdravotnictví, jak nám vzkazuje na krabièkách, ale nikterak ho to nevzrušuje zøejmì?, nebo nechává cigarety dále distribuovat jako potraviny - v této souvislosti je smutné zjištìní, že žvýkací tabák neprojde.. Cigarety ano? Proè tedy cigareta není novokuøákùm zakázána, co hledá v kapitole "Potravináøský a tabákový prùmysl?" Co hledá na pultu každých potravin, ve skryté i otevøené reklamì. Dùmyslnou strategií tak asociuje zejména mladým lidem, že ono to zase tak hrozné není, to by to nedali do potravin, to by nekouøil ten a ten... Cigareta bez zábran hledá nové obìti a jejich nárùst u žákù základních i støedních škol je rekordní za poslední desetiletí.

75. 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.shodor.org/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 use this form for questions and comments about this project. The Shodor Education Foundation, Inc.

76. Waclaw Sierpinski - Wikipedia waclaw Franciszek sierpinski (ur. 14 marca 1882 r. w Warszawie zm. 21 pazdziernika1969 r. w Warszawie) - polski matematyk. http://pl.wikipedia.org/wiki/Wacław_Sierpiński

77. Polish School Of Mathematics Any Examination Of Twentieth Century As noted above, Janiszewski, along with Stefan Banach, waclaw sierpinski, andStanislaw Zaremba were all instrumental in the development of the Polish http://www.math.wfu.edu/~kuz/Stamps/PolishSchool/PolishSchool.htm

Polish School of Mathematics Any examination of twentieth century mathematics shows the surprising depth, originality and quantity of Polish contributions to the discipline. Similarly, any list of important twentieth century mathematicians contains Polish names in a frequency out of proportion to the size of the country. How did such creativity and mathematical influence develop in a country that had little tradition in research, that was partitioned under foreign domination from 1795 until the end of World War I, and whose educational institutions were suppressed by foreign powers. Surprisingly, it was planned! What was to become known as the Polish School of Mathematics was established following a plan proposed by Zygmunt Janiszewski. Poradnik dla Samoukow (Guidebooks for Self-Instruction) . These were designed to get around the Russian and German e ducational restrictions and were written by prominent mathematicians including Janiszewski, Sierpinski, and Zaremba; they covered topics such as series, differential and integral equations, and topology. Another series it supported was ; the first issue (1917) contained two articles which were to be very important to the establishment of the Polish School of Mathematics.

78. INDEX OF NAMES sierpinski, waclaw (Poland, 18821969) and nested curves, 934 Silverman,Brian (Canada, 1957- ) in Preface, xiii and WireWorld CA, 1117 http://www.wolframscience.com/nksonline/index/names/p-s.html?SearchIndex=Post, E

79. Sh-So sierpinski, waclaw (Poland, 18821969) and nested curves, 934 sierpinski carpet,933 sierpinski pattern and 2D substitution system, 187 http://www.wolframscience.com/nksonline/index/sh-so.html?SearchIndex=Shift regis

80. Il Triangolo Di Waclaw Sierpinski - Calcolo Del Perimetro - Caos E Oggetti Fratt Translate this page COME VARIA IL PERIMETRO DEL TRIANGOLO DI sierpinski? Macchina per funzioniche calcola il perimetro del triangolo di sierpinski fino al passo 20 http://www.webfract.it/FRATTALI/perimetroEs.htm

COME VARIA IL PERIMETRO DEL TRIANGOLO DI SIERPINSKI? Precedente Successivo Passo Situazione Perimetro Rapporto del perimetro al precedente Macchina per funzioni che calcola il perimetro del triangolo di Sierpinski fino al passo 20 1 triangolo di lato 1 Scegli il numero di passi Perimetro? Il perimetro del triangolo diventa ogni volta i 3/2 del precedente, infatti i triangoli si triplicano mentre il loro lato si dimezza. Possiamo dunque affermare che, al crescere del numero dei passi, anche il perimetro crescerà indefinitamente: esso tende ad infinito quando anche il numero di passi tende ad infinito. 3 triangoli di lato 1/2 9 triangoli di lato 1/4 27 triangoli di lato 1/8 di sicuro sarà 3/2 k k triangoli di lato -k k -k k+1 k di sicuro sarà 3/2 TORNA INDIETRO www.webfract.it

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