Sierpinski triangle

The Sierpinski triangle, also called the Sierpinski gasket, is a fractal, named after Wacław Sierpiński who described it in 1916. Originally constructed as a curve, this is one of the basic examples of self-similar sets. Image File history File links Download high-resolution version (1122x949, 3 KB) Summary Sierpinski Triangle drawn by an original recursive Java method I have written myself. ... Image File history File links Download high-resolution version (1122x949, 3 KB) Summary Sierpinski Triangle drawn by an original recursive Java method I have written myself. ... The boundary of the Mandelbrot set is a famous example of a fractal. ... WacÅ‚aw Franciszek SierpiÅ„ski (March 14, 1882 — October 21, 1969), a Polish mathematician, was born and died in Warsaw. ... 1916 (MCMXVI) was a leap year starting on Saturday (link will take you to calendar). ...

Construction

An algorithm for obtaining arbitrarily close approximations to the Sierpinski triangle is as follows:

Image File history File links Download high-resolution version (2480x480, 10 KB) Sierpinski triangle, the evolution in five iterations. ...

1. Start with any triangle in a plane. The canonical Sierpinski triangle uses an equilateral triangle with a base parallel to the horizontal axis (first image).
2. Shrink the triangle by ½, make two copies, and position the three shrunken triangles 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 Sierpinski gasket. Michael Barnsley used an image of a fish to illustrate this in his paper V-variable fractals and superfractals (PDF). For alternate meanings, such as the musical instrument, see triangle (disambiguation). ... Michael Barnsley is the researcher and entrepreneur who has worked on fractal compression; he holds several patents on the technology. ...

Image File history File links Sierpinski gasket starting from a square Made by SGBailey; no copyright. ...

The actual fractal is what would be obtained after an infinite number of iterations. More formally, one describes it in terms of functions on closed sets of points. If we let d_a note the dilation by a factor of ½ about a point a, then the Sierpinski triangle with corners a, b, and c is the fixed set of the transformation d_a U d_b U d_c.

This is an attractive fixed set, so that when the operation is applied to any other set repeatedly, the images converge on the Sierpinski triangle. This is what is happening with the triangle above, but any other set would suffice.

If one takes a point and applies each of the transformations d_a, d_b, and d_c to it randomly, the resulting points will be dense in the Sierpinski triangle, so the following algorithm will again generate arbitrarily close approximations to it:

Start by labelling p₁, p₂ and p₃ as the corners of the Sierpinski triangle, and a random point v₁. Set v_n+1 = ½ ( v_n + pr_n ), where r_n is a random number 1, 2 or 3. Draw the points v₁ to v_∞. If the first point v₁ was a point on the Sierpinski triangle, then all the points v_n lie on the Sierpinski triangle. If the first point v₁ to lie within the perimeter of the triangle is not a point on the Sierpinski triangle, none of the points v_n will lie on the Sierpinski triangle, however they will converge on the triangle. If v₁ is outside the triangle, the only way v_n will land on the actual triangle, is if v_n is on what would be part of the triangle, if the triangle was infinitely large.

Or simpler:

1. Take 3 points in a plane, start at one of the points.
2. Randomly select any point and move half the distance from that point to any of the 3 vertex points. Plot the current position.
3. Repeat from step 2.

Note: This method is also called the Chaos Game. You can start from any point outside or inside the triangle, and it would eventually form the Siepinski Gasket with a few leftover points. However, it is not a bright idea to do this with pencil and paper. It would require at least 7,500 points before even a brief outline is formed.

Or using an Iterated function system

An alternative way of computing the Sierpinski triangle uses an Iterated function system and starts by a point in the origin (x₀=0, y₀=0) and then the new points are iteratively computed by randomly applying (with equal probability) one of the following 3 coordinate transformations (using the so called chaos game): Menger sponge, created by using IFS. Iterated function systems or IFSs, are a kind of fractal which were conceived in their present form by John Hutchinson in 1981 and popularized by Michael Barnsleys book Fractals Everywhere. ... The chaos game or chaosgame is a means of creating a fractal, using a polygon and a random point inside it. ...

Sierpinski triangle using IFS

x_n+1=0.5 x_n
y_n+1=0,5 y_n  ; a half-size copy
when this coordinate transformation is used, the point is drawn in yellow in the figure Image File history File links Serpinski1. ... Image File history File links Serpinski1. ...

x_n+1=0.5 x_n+0,5
y_n+1=0,5 y_n+0,5 ; a half-size copy shifted right and up
when this coordinate transformation is used, the point is drawn in red

x_n+1=0.5 x_n+1
y_n+1=0,5 y_n  ; a half-size copy doubled shifted to the right
when this coordinate transformation is used, the point is drawn in blue

Or using an L-system — The Sierpinski triangle drawn using an L-system. See L-system for information on Lindenmayer systems. ...

Other means — The Sierpinski triangle also appears in certain cellular automata, including those relating to Conway's Game of Life. The automaton "12/1" when applied to a single cell will generate four approximations of the Sierpinksi triangle. A cellular automaton (plural: cellular automata) is a discrete model studied in computability theory and mathematics. ... Gospers Glider Gun creating gliders. The Game of Life is a cellular automaton devised by the British mathematician John Horton Conway in 1970. ...

Properties

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. In mathematics, the Hausdorff dimension is an extended non-negative real number (that is a number in the closed infinite interval [0, ∞]) associated to any metric space . ...

If one takes Pascal's triangle with 2ⁿ rows and colors the even numbers white, and the odd numbers black, the result is an approximation to the Sierpinski triangle. 1 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1 The first six rows of Pascals triangle In mathematics, Pascals triangle is a geometric arrangement of the binomial coefficients in a triangle. ...

The area of a Sierpinski triangle is zero (in Lebesgue measure). This can be seen from the infinite iteration, where we remove 25% of the area left at the previous iteration. In mathematics, the Lebesgue measure is the standard way of assigning a volume to subsets of Euclidean space. ...

Analogs in higher dimension

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. A tetrahedron (plural: tetrahedra) is a polyhedron composed of four triangular faces, three of which meet at each vertex. ...

A Sierpinski pyramid and its 'inverse'

Image File history File links Download high resolution version (1920x1200, 2613 KB) Summary Rendering of a Sierpinski pyramid and its inverse, using a radiosity renderer. ... Image File history File links Download high resolution version (1920x1200, 2613 KB) Summary Rendering of a Sierpinski pyramid and its inverse, using a radiosity renderer. ...

Self-assembly with DNA

Researchers in Erik Winfree's lab at the California Institute of Technology have also constructed self-assembling Sierpinski triangles using DNA tiles (Rothemund et al. 2004). The California Institute of Technology (commonly referred to as Caltech)[1] is a private, coeducational university located in Pasadena, California, in the United States. ...

Computer program

Computer programs to draw Sierpinski triangles are usually done recursively in programming languages that permit explicit recursion such as Java. A Sierpinski triangle —a confined recursion of triangles to form a geometric lattice. ... A programming language is an artificial language that can be used to control the behavior of a machine (often a computer). ... Java is an object-oriented programming language developed by James Gosling and colleagues at Sun Microsystems in the early 1990s. ...

 import java.awt.*; import java.applet.*; public class SierpinskiTriangle extends Applet { private Graphics g; private int dMin=4; // limit to recursion in pixels public void paint(Graphics g) { this.g = g; int d = 1024; // basis (width of the triangle) int x0 = 50; // distance from the left int y0 = 50; // distance from the top int h = (int)(d*Math.sqrt(3)/2); // height // so: suitable for an equilateral triangle int xA=x0, yA=y0+h; // (bottom-left) int xB=x0+d, yB=y0+h; // (bottom-right) int xC=x0+d/2, yC=y0; // equilateral triangle (top-center) int[] x = { xA, xB, xC }; int[] y = { yA, yB, yC }; drawSierpinskiTriangle( x, y, d/2 ); // start recursion } private void drawSierpinskiTriangle ( int[] x, int[] y, int d ) { if (d<=dMin) g.fillPolygon ( x, y, 3 ); // bottom of the recursion else { // centers of the sides: int xMc = (x[0]+x[1])/2, yMc = (y[0]+y[1])/2; int xMb = (x[0]+x[2])/2, yMb = (y[0]+y[2])/2; int xMa = (x[1]+x[2])/2, yMa = (y[1]+y[2])/2; int[] xNew1 = { x[0], xMc, xMb }; int[] yNew1 = { y[0], yMc, yMb }; drawSierpinskiTriangle ( xNew1, yNew1, d/2 ); // recursion int[] xNew2 = { x[1], xMc, xMa }; int[] yNew2 = { y[1], yMc, yMa }; drawSierpinskiTriangle ( xNew2, yNew2, d/2 ); // recursion int[] xNew3 = { x[2], xMb, xMa }; int[] yNew3 = { y[2], yMb, yMa }; drawSierpinskiTriangle ( xNew3, yNew3, d/2 ); // recursion } } } 

Another implementation doesn't use recursion and draws the triangle in a rotated maner. But is easier to implement.

 // Assumes: // - plot plots a pixel on a surface // - size is a power of two // - & is a bitwise and operator // for(int x = 0; x < size; x++) for(int y = 0; y < size; y++) if(x & y) plot(x, y, background) else plot(x, y, foreground) 

See also

• List of fractals by Hausdorff dimension
• Sierpinski carpet
• Pascal's triangle

A fractal is a geometric object whose Hausdorff dimension (δ) strictly exceeds its topological dimension. ... The Sierpinski carpet is a plane fractal first described by Wacław Sierpiński. ... 1 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1 The first six rows of Pascals triangle In mathematics, Pascals triangle is a geometric arrangement of the binomial coefficients in a triangle. ...

References

• Weisstein, Eric W., Sierpinski Sieve at MathWorld.
• Paul W. K. Rothemund, Nick Papadakis, and Erik Winfree, Algorithmic Self-Assembly of DNA Sierpinski Triangles, PLoS Biology, volume 2, issue 12, 2004.

Wikimedia Commons has media related to:

Sierpinski triangle

• Sierpinski Gasket by Trema Removal at cut-the-knot
• Sierpinski Gasket and Tower of Hanoi at cut-the-knot
• Article explaining Sierpinski's Triangle created with a bitwise XOR (example program in Macromedia Flash ActionScript)
• Article explaining Sierpinski's Triangle created with the Chaos Game (example program in Macromedia Flash ActionScript)
• Sierpinski Triangle and Machu Picchu. Fractal illustration with animation and sound.
• VisualBots - Freeware multi-agent simulator in Microsoft Excel. Sample programs include Sierpinski Triangle.

MathWorld is an online mathematics reference work, sponsored by Wolfram Research Inc. ... Image File history File links Commons-logo. ... Wikimedia Commons logo by Reid Beels The Wikimedia Commons (also called Commons or Wikicommons) is a repository of free content images, sound and other multimedia files. ... This article is being considered for deletion in accordance with Wikipedias deletion policy. ... This article is being considered for deletion in accordance with Wikipedias deletion policy. ...

External links

• IFS Fractal fern and Sierpinski triangle - JAVA applet
• Contains a section where the Sierpinski triangle can be seen step by step -- Shockwave