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# Topic: Sierpinski curve

###### In the News (Wed 21 Aug 19)

 Wacław Sierpiński - Wikipedia, the free encyclopedia Sierpinski began to study set theory and, in 1909, he gave the first ever lecture course devoted entirely to the subject. Sierpinski was awarded the scientific prize of the first degree in 1949. Waclaw Sierpinski is interred in the Powązki Cemetery, Warsaw, Poland. en.wikipedia.org /wiki/Waclaw_Sierpinski   (862 words)

 Fractals: Sierpinski Curve The Sierpinski curve, named from the polish mathematician Waclaw Sierpinski who originally devised it around 1912, is much less known than the other fractal objects created by Sierpinski and his co-workers as the Sierpinski gasket or the Sierpinski Carpet. The Sierpinski curve also share the very interesting property of the most fractals: its area converges rapidly to a finite limit while the total length of the segments that composed that curve have no limit. This was a period of Russian occupation of Poland and despite the difficulties, Sierpinski entered the Department of Mathematics and Physics of the University of Warsaw in 1899. users.skynet.be /TGMDev/curvesierpinski.htm   (997 words)

 Space-Filling Curves   (Site not responding. Last check: 2007-10-22) For example, the name Hilbert space-filling curve should properly be used only for the limit curve reached as the level parameter of the Hilbert curves tends to infinity. The Sierpinski group of curves use isosceles right-angled triangles as cells. The Sierpinski_C curve and the Sierpinski curve are, in a sense, complements of each other: the shape of the areas excluded by one curve is the shape included by the other. www.cs.utexas.edu /users/vbb/misc/sfc/Oindex.html   (511 words)

 Waclaw Sierpinski   (Site not responding. Last check: 2007-10-22) Sierpinski graduated in 1904 and worked for a while as a school teacher of mathematics and physics in a girls school in Warsaw. The length of the curve is infinity, while the area enclosed by it is 5/12 that of the square. Sierpinski continued working in the "Underground Warsaw University" while his official job was a clerk in the council offices in Warsaw. www.stetson.edu /~efriedma/periodictable/html/Si.html   (611 words)

 Sierpinski curve   (Site not responding. Last check: 2007-10-22) The Sierpinski curve is a base motif fractal where the base is a square. The curve is the only plane locally connected one-dimensional continuum S such that the boundary of each complementary domain of S is a simple closed curve and no two of these complementary domain boundaries intersect. The curve is a two-dimensional generalization of the Cantor set. www.2dcurves.com /fractal/fractals.html   (241 words)

 Sierpinski Curve   (Site not responding. Last check: 2007-10-22) While the Hilbert curve is an open curve, the Sierpinski curve is closed; it bounds an area that is 5/12 that of the square it fills. The Sierpinski curve consists of four open curves linked together by line segments located in the outer corners. Like the Hilbert curve, the Sierpinski curve is also the limit of an infinite series of curves. home.comcast.net /~davebowser/fractals/sierpinski.html   (383 words)

 Sierpinski   (Site not responding. Last check: 2007-10-22) Sierpinski was awarded the gold medal in the competition for his dissertation. Sierpinski was lucky for the lector changed the mark on his Russian language course to 'good' so that he could take his degree. Sierpinski spoke of the tragic events of the war during a lecture he gave at the Jagiellonian University in Krakóv in 1945 (see [13]). www-gap.dcs.st-and.ac.uk /~history/Mathematicians/Sierpinski.html   (1772 words)

 Sierpinski Carpet The Sierpinski carpet is the intersection of all the sets in this sequence, that is, the set of points that remain after this construction is repeated infintely often. Sierpinski used the carpet to catalogue all compact one-dimensional objects in the plane (from a topological point of view). So the Sierpinski carpet was actually invented by Stefan Mazurkiewicz, who in 1913 wrote his Ph.D. thesis under the supervision of Sierpinski on curves filling a square. ecademy.agnesscott.edu /~lriddle/ifs/carpet/carpet.htm   (523 words)

 Sierpinski Gasket The Sierpinski gasket, also known as the Sierpinski triangle, is the intersection of all the sets in this sequence, that is, the set of points that remain after this construction is repeated infinitely often. The Sierpinski gasket is also referred to as the Sierpinski triangle or as the Sierpinski triangle curve. It apparently was Mandelbrot who first gave it the name "Sierpinski's gasket." Sierpinski described the construction to give an example of "a curve simultaneously Cantorian and Jordanian, of which every point is a point of ramification." Basically, this means that it is a curve that crosses itself at every point. ecademy.agnesscott.edu /~lriddle/ifs/siertri/siertri.htm   (818 words)

 Recent Papers of Robert L. Devaney These halos are Sierpinski holes in which the corresponding maps have Julia sets that are Sierpinski curves. These Sierpinski curves have complementary domains that consist not only of the basin of attraction of infinity, but also of the basins of other finite attracting cycles. These Julia sets are generalized Sierpinski gaskets and are produced when the critical orbits for these maps land on repelling periodic points in the basin of infinity. math.bu.edu /people/bob/papers.html   (2277 words)

 The Logistics Institute at Georgia Tech - Research   (Site not responding. Last check: 2007-10-22) A famous spacefilling curve is that due to Sierpinski, which is formed by repeatedly copying and shrinking a simple pattern (the convoluted tour in Figure 1). A useful property of a spacefilling curve is that it tends to visit all the points in a region once it has entered that region. To accompany this is a table of Sierpinski indices of the points of a 100 x 100 grid [pdf format, 22 pages], with which you can set up your own routing system in an afternoon. www.tli.gatech.edu /research/casestudies/spacefilling_curves   (713 words)

 A Universal Separable Metric Space Based on the Triangular Sierpinski Curve   (Site not responding. Last check: 2007-10-22) Sierpinski has constructed two significant curves at the beginning of this century. A year later he constructed his rectangular curve and proved that is universal for planar one-dimensional spaces. The rectangular curve is a particular case of today called Menger spaces. www.pmf.ukim.edu.mk /mathematics/ivansic.htm   (186 words)

 Fractals: Sierpinski Objects The curve on the left was drawn with the geometric method. Intuitively, the length of the Sierpinski gasket is the total of the length of all the segments required to draw the object. As the middle one is left unpainted, the total area of the Sierpinski curve after the first iteration is 3/4 of the original area. users.skynet.be /TGMDev/curvesierpinskiobj.htm   (1475 words)

 plot79_d/dem81.html   (Site not responding. Last check: 2007-10-22) C$C$ Apart from its geometrically interesting shape, tending C$toward space filling, the principle interest of the C$ Sierpinski curve here is that it can be made arbitrarily C$long by simply increasing the order, and it therefore may C$ serve as a useful test of plotter speed. C$C$ The Sierpinski curves are closed, and a curve of order N C$has 4**(N+1)+1 vertices. The curves are also excellent tests of polygon C$ edge fill algorithms, which may not be limited by internal C$stacks, but nevertheless can fail to handle complex C$ polygons, or polygons with colinear edges. www.math.utah.edu /software/plot79/plot79_d/dem81.html   (203 words)

 Menger sponge   (Site not responding. Last check: 2007-10-22) The curve is the most popular creation of the mathematician Karl Menger, while working on dimension theory. The curve has as alternative names: the Menger universal curve, or the Sierpinski sponge. The curve is a three-dimensional variant on the Cantor set and the Sierpinski curve. www.2dcurves.com /3d/3dm.html   (178 words)

 The CTK Exchange Forums Since the sierpinski gasket is fractal, such a function could only (as far as I know) be defined as a limit. It is evident that f is defined on the interval <0, 1> and that the image of f is the sierpinski gasket. Handily, your definition of the sierpinski gasket in terms of areal co-ordinates makes it clear from the rules of the game that f(x) is always an element of the sierpinski gasket. www.cut-the-knot.com /htdocs/dcforum/DCForumID5/282.shtml   (1432 words)

 An English-Persian Dictionary of Algorithms and Data Structures   (Site not responding. Last check: 2007-10-22) Definition: A fractal whose envelope is an equilateral triangle and which is composed of three half-sized Sierpinski triangles. Note: This definition leads to a recursive algorithm to draw (an approximation of) a Sierpinski triangle. The Sierpinski triangle S on these points is the set that satisfies the following conditions. ce.sharif.edu /~dic-ads/d.php?r=Sierpinski+triangle.4   (196 words)

 The Sierpinski curve: a lesson in debugging and conversion. The Sierpinski curve: a lesson in debugging and conversion. Sometime in the 1700's, mathematicians described the word "curve" more precisely than had been done previously by defining a curve as the loci of points that satisfy equations that are continuous functions. Peano's curve is a legitimate diagram of a continuous function, yet nowhere on it can a unique tangent be drawn because at no instant can one specify the direction in which the point is moving. www.atarimagazines.com /creative/v10n7/148_The_Sierpinski_curve_a_l.php   (1774 words)

 Arlo Caine's Web Page / Mathematics / Limits with ...   (Site not responding. Last check: 2007-10-22) 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. In fact, this sequence of drawings does have a limit, in a technical sense, and that limit is called "von Koch's Curve." What's interesting, is that if you arrange 3 copies of the curve along the edges of an equilateral triangle, you get the figure at left. The limit of this sequence is called Sierpinski's Gasket, named for Waclaw Sierpinski (1882-1969), a polish mathematician who made great contributions to the foundations of mathematics. math.arizona.edu /~caine/chaos.html   (1180 words)

 Hyperbolic Groups With Low Dimensional Boundary - Kapovich, Kleiner (ResearchIndex)   (Site not responding. Last check: 2007-10-22) When @1G is a Sierpinski carpet we show that G is a quasiconvex subgroup of a 3-dimensional hyperbolic Poincar'e duality group. 3 Topological characterization of the Sierpinski curve (context) - Whyburn 2 A characterization of the universal curve and a proof of its.. citeseer.ist.psu.edu /kapovich00hyperbolic.html   (827 words)

 CVM 1.1 (VSFCF): The Peano Curve   (Site not responding. Last check: 2007-10-22) Peano was the first to describe a space-filling curve ([Pe]). This curve can be described with an IFS of nine functions as described by Sagan ([Sag]). By adding a shrinking parameter and connecting segments, as we did for the Hilbert curve, we get Figure 10 (from which one can also deduce the mappings in the IFS). www.maa.org /cvm/1998/01/vsfcf/article/sect8/peano.html   (93 words)

 Home page for spacefilling curves and applications This application was communicated to us by scientists from TRW Systems, an SDI contractor, who chose the spacefilling curve heuristic over alternatives because it was well-analyzed, parallelizable, and could run on a computer that was boostable to orbit. This tour was induced by the Sierpinski spacefilling curve in less than a second and is about 1/3 again as long as the shortest possible. Vertex-labelling algorithms for the Hilbert spacefilling curve by J. Bartholdi and P. Goldsman (2000). www.isye.gatech.edu /people/faculty/John_Bartholdi/mow/mow.html   (1087 words)

 Sierpinski Triangle   (Site not responding. Last check: 2007-10-22) 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. As early as 1915, Sierpinski described a "gasket" or a "triangle" with repeated and proportionally reduced areas. Today these shapes are widely known as "fractals." Sierpinski's triangles would later emerge to be among the most recognizable shapes or patterns in all computer graphics. curvebank.calstatela.edu /sierpinski/sierpinski.htm   (218 words)

 National Curve Bank: A Math Archive   (Site not responding. Last check: 2007-10-22) The National Curve Bank is a resource for students of mathematics. We also include geometrical, algebraic, and historical aspects of curves, the kinds of attributes that make the mathematics special and enrich classroom learning. Please see "Submit Your Curve" on the left for details. curvebank.calstatela.edu /home/home.htm   (109 words)

 Fractal Space Filling Curves The shapes these curves fill can be subdivided into a pattern of tiles, where each tile is similar in shape to the whole. In this example, each more complicated version of the curve is made by replacing each colored portion with a copy of the whole curve (perhaps mirrored or rotated). The Sierpinski curve is based on a triangle divided into two tiles. www.seanet.com /~garyteachout/fill.html   (527 words)

 CSC 115 Assignment 3 Fall 2002 The curves in the assignment belong to the set of fractal drawings. Add the Cross Curve as another curve on the frame that is instantiated with SpaceFill (you also need to extend the user interface to include Cross as an option). These pages illustrate the recursive construction rules for the curves we are using in this project: (please note that the graphic y-axis points down, as opposed to the mathematical y-axis, which points up). csr.uvic.ca /~mstorey/teaching/csc115/assignments/assignment_3.html   (469 words)

 CVM 1.1 (VSFCF): Cover Image   (Site not responding. Last check: 2007-10-22) This image represents the graph of a curve that passes through a fractal Cantor set of dimension 1.5. It is just one of sequence of curves whose dimensions change continuously from 1 to 2, with the dimension 2 curve forming the classical Sierpinski-Knopp space-filling curve. This is described more fully in the article Visualizing Space-Filling Curves with Fractals (As Limits of Curves of Continuously Varying Dimension), specifically in the section on the Sierpinski-Knopp curve www.geom.uiuc.edu /~dpvc/CVM/1998/01/cover/vsfcf.html   (114 words)

 Atlas: Sierpinski Curve Julia Sets for Singularly Perturbed Rational Maps by Daniel M. Look   (Site not responding. Last check: 2007-10-22) Sierpinski curves are interesting sets in that all Sierpinski curves are known to be homeomorphic and these sets contain a homeomorphic copy of every one-dimensional planar continuum. and show that for infinitely many choices of \lambda the Julia set is a Sierpinski curve. The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # camc-30. atlas-conferences.com /cgi-bin/abstract/camc-30   (121 words)

 The Fractal Nature of Semantics This is followed by a pattern that resembles a butterfly: this is the Lorenz Attractor for modeling weather forecasts. Query a system and construct several precision-recall curves using a variable size consisting of the top 10, 20,...100,... Examine the effect of N on the precision-recall curves. www.miislita.com /fractals/fractal.html   (1865 words)

 CVM 1.1 (VSFCF): Software and Source Codes Approximations to Hilbert's curve of steadily increasing dimension Fractal versions of 3-dimensional approximations to a Hilbert curve filling a cube. Fractal versions of 3-dimensional approximations to a Peano curve filling a cube. www.maa.org /cvm/1998/01/vsfcf/article/sect13/software.html   (280 words)

 Atlas: Buried Sierpinski Curve Julia Sets by Daniel M. Look   (Site not responding. Last check: 2007-10-22) = 0 there are infinitely many parameter values for which the Julia set is a Sierpinski curve on which the dynamics are distinct. In each of the above cases where the Julia set is a Sierpinski curve, the complementary domains (or the Fatou components) are always preimages of the immediate basin of attraction of The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # capc-06. atlas-conferences.com /cgi-bin/abstract/capc-06   (214 words)

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