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# Topic: Sierpinski triangle

###### In the News (Sun 21 Jul 19)

 The Sierpinski triangle. The Sierpinski triangle S may also be constructed using a deterministic rather than a random algorithm. From each remaining triangle we remove the "middle" leaving behind three smaller triangles each of which has dimensions one-half of those of the parent triangle (and one-fourth of the original triangle). It is easy to check that the dimensions of the triangles that remain after the Nth iteration are exactly 1/2^N of the original dimensions. math.bu.edu /DYSYS/chaos-game/node2.html   (155 words)

 [ wu :: fractals | sierpinski ] 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. www.ocf.berkeley.edu /~wwu/fractals/sierpinski.html   (416 words)

 [No title] The original Sierpinski gasket, also known as Sierpinski triangle, is a point set obtained as a limiting configuration when an infinite sequence of inverted equilateral triangles that decrease in size steadily according to a certain rule is removed from an equilateral triangle in standard position. SIERPINSKI GASKET The classical fractal known as the Sierpinski gasket or Sierpinski triangle was first introduced in 1916 by the great Polish mathematician, Waclaw Sierpinski (1882-1969). The nodes of the tree are the centroids of these triangles and the branches of the tree grow from the nodes of one generation to those of the next, as shown in Figure 3 for the first three generations. www.csupomona.edu /~jis/1996/Glaser.doc   (5244 words)

 Sierpinski triangle - Wikipedia, the free encyclopedia The Sierpinski triangle has Hausdorff dimension log(3)/log(2) ≈ 1.585, which follows from the fact that it is a union of three copies of itself, each scaled by a factor of ½. The area of a Sierpinski triangle is zero (in Lebesgue measure). The tetrix is the three-dimensional analog of the Sierpinski triangle, formed by repeatedly shrinking a regular tetrahedron to one half its original height, putting together four copies of this tetrahedron with corners touching, and then repeating the process. en.wikipedia.org /wiki/Sierpinski_triangle   (1020 words)

 Sierpinski Gasket   (Site not responding. Last check: 2007-10-24) 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)

 Heidi Sandbulte Sierpinski’s Triangle is al= so known as the Sierpinski Gasket. If we break our triangle down into steps, we find that the beginning triangle is broken into four parts with the middle one removed. This means tha= t we have a triangle with an infinite perimeter and a finite area. www.thecoo.edu /~hesand/GeometryProject.mht   (709 words)

 Binary Description of Sierpinski Triangle   (Site not responding. Last check: 2007-10-24) In each of the remaining three right triangles, remove the triangle formed by the midpoints of the three sides. We position the triangle in the first quadrant so that the two sides of unit length lie along the x-axis and y-axis. The Sierpinski Gasket is the set of all pairs (x, y) such that for each position in the binary representation, at least one of the corresponding binary digits of x and y in that position is 0, that is, ecademy.agnesscott.edu /~lriddle/ifs/siertri/binary.htm   (603 words)

 Sierpinski Triangles A Sierpinski triangle is a recursively defined structure, since each of the three outer triangles formed by joining the midpoints is itself a Sierpinski triangle. So a Sierpinski triangle consisting of a single triangle has depth 0; when a triangle is drawn inside of it, the resulting Sierpinski triangle has depth 1; when the three outside triangles have triangles drawn inside of them, the resulting triangle is depth 2, and so on. The sierpinski method will then check to see if the desired depth has been exceeded (the depth can be a constant), and if not, draw the triangle, then call itself recursively to drawn the three Sierpinski triangles that will be embedded. www.lausd.k12.ca.us /Hollywood_HS/ap/lab/ch11/Sierpinski.html   (534 words)

 Fractals: Sierpinski Objects Intuitively, the length of the Sierpinski gasket is the total of the length of all the segments required to draw the object. On the next iteration, each triangle gives three smaller triangles having a perimeter half of that of their parent triangle, so that the total length is then equal to 9/4 of the original perimeter. 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.swing.be /TGMSoft/curvesierpinskiobj.htm   (1475 words)

 Cynthia Lanius' Fractals Unit: The Sierpinski Triangle Connect the midpoints of the sides and shade the triangle in the center as before. Notice the three small triangles that also need to be shaded out in each of the three triangles on each corner - three more holes. These students used Technology to draw the Sierpinski Triangle. math.rice.edu /~lanius/fractals   (189 words)

 Sierpinski's Triangle Fractal   (Site not responding. Last check: 2007-10-24) The Sierpinski triangle fractal is made by drawing three lines that connect the midpoints of the triangle's sides. When the sides of each triangle in an iteration are divided by two, the smaller resulting triangles are magnified by 1/n. As the fractal is further divided into smaller triangles the blue area of the initial triangle reduces. www.spots.ab.ca /~belfroy/mathematics/TriangleFractalText.html   (335 words)

 The Geometry Junkyard: Sierpinski Tetrahedra The Sierpinski Tetrahedron has Hausdorff dimension two, so maybe it's not really a fractal in the "fractional dimension" sense of the word. Rainbow Sierpinski tetrahedron by Aécio de Féo Flora Neto. Sierpinski triangle reptile based on a complex binary number system, R. Gosper. www.ics.uci.edu /~eppstein/junkyard/sierpinski.html   (493 words)

 Sierpinski gasket A fractal, also known as the Sierpinski Triangle or Sierpinski Sieve after its inventor Waclaw Sierpinski. It is produced by the following set of rules: (1) start with any triangle in a plane; (2) shrink the triangle by 1/2, make three copies, and translate them so that each triangle touches the two other triangles at a corner; (3) repeat step 2 ad infinitum. The Gasket has a Hausdorff dimension of log 3/log 2 = 1.585..., which follows from the fact that it is a union of three copies of itself, each scaled by a factor of 1/2. www.daviddarling.info /encyclopedia/S/Sierpinski_gasket.html   (292 words)

 Sierpinski triangle Summary The canonical Sierpinski triangle uses an equilateral triangle with a base parallel to the horizontal axis (first image). The Sierpinski triangle has Hausdorff dimension log(3)/log(2) ≈ 1.585, which follows from the fact that it is a union of three copies of itself, each scaled by a factor of 1/2. Computer programs to draw Sierpinski triangles are usually done recursively in programming languages that permit explicit recursion such as Java. www.bookrags.com /Sierpinski_triangle   (1681 words)

 Programmed DNA forms fractal TRN 040605 A Sierpinski triangle is a type of crystal, or structure that regularly repeats. The DNA Sierpinski triangles show that there is no theoretical barrier to using molecular self-assembly to carry out any kind of computing and nanoscale fabrication, according to Winfree. In the case of a Sierpinski pattern, the molecules are directing the process of self-assembly, he said. www.trnmag.com /Stories/2005/040605/Programmed_DNA_forms_fractal_040605.html   (990 words)

 Infinity the Area of a Sierpinski Gasket Instead of being concerned with the actual area of the equilateral triangles, it is possible to assign an arbitrary value to the area of the large white one and compute the area of the fl in terms of that value. To make the first fl triangle that will be glued on the large white triangle (the one that connects the midpoints of its sides), a fl triangle the same size as the white one is folded in a way that actually makes 4 smaller fl triangles. The fraction 1/d, (where d is the denominator of each term) is the measure of the area of one triangle in relation to the large white triangle whose area was taken to be 1. www.c3.lanl.gov /mega-math/new/sierpins/infarea.html   (1156 words)

 NetLogo Models Library: Sierpinski Simple Sierpinski was a professor at Lvov and Warsaw. The Sierpinski tree is closely related to the class of fractals called Sierpinski Carpets which includes the famous Sierpinski Triangle or as it is usually called The Sierpinski Gasket. However connectedness is apparent from the way Sierpinski tree is generated; at each iteration the set is connected. ccl.northwestern.edu /netlogo/models/SierpinskiSimple   (377 words)

 Sierpinski Tautology Map In material implication the Sierpinski triangle shifts to the lower left as a value pattern for tautology. The argument below regarding the appearance of the Sierpinski triangle applies to a full continuum as well as the envisaged subsets, and will be valid both for the full propositional calculus and for the envisaged approximations to it. It is well known that the points constitutive of the Sierpinski gasket within a continuous unit square are infinitely many, but nonetheless 'very few' in the sense that a random selection of points has a probability approaching zero of hitting such a point. www.sunysb.edu /philosophy/fractal/Sierpins.html   (2671 words)

 Sierpinski Gasket If the entry in pascals triangle is odd then it is part of the gasket otherwise it is not part of the gasket. Instead of removing the central third of a triangle, the central square piece is removed from a square sliced into thirds horizontally and vertically. The Menger sponge is the 3D equivalent of the Sierpinski carpet which is in turn the 2D equivalent of the 1D Cantor set. local.wasp.uwa.edu.au /~pbourke/fractals/gasket   (2172 words)

 Fractals   (Site not responding. Last check: 2007-10-24) If the given triangle is still large enough, the function must draw the triangle and possibly some nested ones. But, each recursive step subdivides the triangle so that the sum of its sides is only half of the given triangle. The original triangle is shown on the left; the desired curve appears on the right. www.htdp.org /2001-09-22/Book/node143.htm   (892 words)

 Sierpinski Gasket and Tower of Hanoi As was discovered by Ian Stewart, puz(Tower of Hanoi) has a surprising relationship to the Sierpinski gasket (also known as the Sierpinski triangle) and, therefore, to Pascal's triangle. In the Tower of Hanoi puzzle, disks stacked on one peg are to be moved to another with, perhaps an intermediate stop at a third, auxiliary peg. In terms of Pascal's triangle, puz(N) combines nodes corresponding to odd entries of Pascal's triangle with the nodes 1 unit (whatever this may mean) apart linked by an edge. www.cut-the-knot.org /triangle/Hanoi.shtml   (1058 words)

 PlanetMath: Sierpinski gasket (alternately the intersection of all these sets) is a Sierpinski gasket, also known as a Sierpinski triangle. Figure: Sierpinski gasket stage 0, a single triangle, and at stage 1, three triangles This is version 19 of Sierpinski gasket, born on 2002-06-02, modified 2005-02-02. planetmath.org /encyclopedia/SierpinskiTriangle.html   (99 words)

 [No title]   (Site not responding. Last check: 2007-10-24) Let's start by giving the algorithm for the Sierpinski triangle by itself and then generalize the algorithm later. The above images speak to one of the advantages of the moving point algorithm for the Sierpinski triangle: speed and uniform fade-in. Now lets look at the Sierpinski triangle we get by merging the three colored corners to get a colorful display of the effect of all three vertices. orion.math.iastate.edu /danwell/Fexplain/ifs1.html   (302 words)

 Exploring Fractals-Area   (Site not responding. Last check: 2007-10-24) This is the area of the Sierpinski Triangle after the 1st iteration. This is the area of the Sierpinski Triangle after the 2nd iteration. Since the area of the Sierpinski Triangle approaches zero, it shows us another way that fractals do not follow the usual rules we have learned for other geometric figures. www.mste.uiuc.edu /courses/ci330ms/eilken/unitarea.HTML   (288 words)

 Sierpinski Pyramid 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 flened, 'filled-in' triangle). 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. The Sierpinski pyramid is a three dimensional version of the one dimensional Sierpinski gasket. www.bearcave.com /dxf/sier.htm   (604 words)

 Sierpinski's Triangle: Mathematics: Stilldreamer Sierpinski’s triangle—also known as the Sierpinski gasket—is a fractal which presents a pattern of nested triangles. The Macromedia Flash animation shown here is a real-time calculation and rendering of the first rows of Sierpinski’s triangle. The method illustrated here is only one of many methods through which we can obtain Sierpinski’s triangle. www.stilldreamer.com /mathematics/sierpinskis_triangle   (224 words)

 All You Ever Wanted to Know About Pascal's Triangle and more For example, in row 3, 1 is the zeroth element, 3 is element number 1, the next three is the 2nd element, and the last 1 is the 3rd element. When all the odd numbers (numbers not divisible by 2) in Pascal's Triangle are filled in (fl) and the rest (the evens) are left blank (white), the recursive Sierpinski Triangle fractal is revealed (see figure at near right), showing yet another pattern in Pascal's Triangle. Go here to download programs that calculate Pascal's Triangle and then use it to create patterns, such as the detailed, right-angle Sierpinski Triangle at the far right. ptri1.tripod.com   (1016 words)

 Mathematicians - Mandelbrot and Sierpinski Then, a triangle is cut out which has it's corners at the midpoint of the original triangle's edges. The shape you end up with, the Sierpinski triangle, has a border which has a length of infinity: every time you repeat the above process you increase the length of the border by one half more than it just did on the last iteration (try it out on paper). But of course, a Sierpinski triangle drawn to five iterations is no real fractal because when you start to zoom in you'll pretty soon see no more selfsimilarity but just big areas of fl and white. mathematica.ludibunda.ch /mathematicians12.html   (901 words)

 [No title] (Technically, you should think of the triangle as an infinite set of points in the plane, namely all the points inside the triangle or on its boundary.) Then, shrink the triangle to half its former width and height, make three copies of it, and arrange them in a triangle. Sure enough, because the triangle is (log_2 3)-dimensional, we can calculate that its measure increases by a factor of (log_2 3) 2 = 3. The parameters for the following code include the x-coordinates of the left and right vertices of the triangle's bottom edge; the y-coordinates of the bottom edge and top vertex; and a number "n" that says how many more iterations need to be done before the recursion bottoms out. www.cs.berkeley.edu /~jrs/4/lec/20   (701 words)

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