Sierpiński triangle 29638 224579455 2008-07-09T14:35:46Z Supuhstar 2636742 /* Analogs in higher dimension */ [[Image:Sierpinski_Triangle.svg|right|thumb|Sierpiński triangle]] The '''Sierpiński triangle''', also called the '''Sierpiński gasket''' or the '''Sierpiński Sieve''', is a [[fractal]] named after [[Wacław Sierpiński]] who described it in [[1915]].<ref>. W. Sierpiński, ''Sur une courbe dont tout point est un point de ramification'', C. R. Acad. Sci. Paris 160(1915) 302-305</ref> Originally constructed as a curve, this is one of the basic examples of [[self-similarity|self-similar]] sets, i.e. it is a mathematically generated pattern that can be reproducible at any magnification or reduction. Comparing the Sierpinski Triangle or the Sierprinski Carpet to equivalent repetitive tiling arrangements, it is evident that similar structures can be built into any rep-tile arrangements. ==Construction== [[image:Sierpinski-zoom4-ani.gif|frame|right|"[[:Image:Sierpinski zoom.gif|Zooming]] in"—it never ends.]] [[image:Animated construction of Sierpinski Triangle.gif|166px|left|thumb|Animated construction <sup>([[:Image:Animated construction of Sierpinski Triangle.gif|Enlarge]])</sup>]] An algorithm for obtaining arbitrarily close approximations to the Sierpiński triangle is as follows: Note: each removed triangle (a ''trema'') is [[topology|topologically]] an [[open set]]. <ref>[http://www.cut-the-knot.org/triangle/Tremas.shtml "Sierpinski Gasket by Trema Removal"]</ref> [[Image:Sierpinski triangle evolution.svg|512px|The evolution of the Sierpiński triangle]] #Start with any triangle in a plane (any closed, bounded region in the plane will actually work). The canonical Sierpiński triangle uses an [[equilateral triangle]] with a base parallel to the horizontal axis (first image). #Shrink the triangle to ½ height and ½ width, make three copies, and position the three shrunken triangles so that each triangle touches the two other triangles at a corner (image 2). Note the emergence of the central hole - because the three shrunken triangles can between them cover only 3/4 of the area of the original. (Holes are an important feature of Sierpiński's Triangle.) #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 Sierpiński gasket. [[Michael Barnsley]] used an image of a fish to illustrate this in his paper "V-variable fractals and superfractals."<ref>[[Michael Barnsley]], ''et al.''{{PDF|[http://www.maths.anu.edu.au/~barnsley/pdfs/V-var_super_fractals.pdf "V-variable fractals and superfractals"]|2.22&nbsp;MB}} </ref> [[Image:Sierpinski triangle evolution square.svg|512px|Iterating from a square]] 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 <math>d_a</math> note the dilation by a factor of ½ about a point a, then the Sierpiński triangle with corners a, b, and c is the fixed set of the transformation <math>d_a</math> U <math>d_b</math> U <math>d_c</math>. This is an attractive fixed set, so that when the operation is applied to any other set repeatedly, the images converge on the Sierpiński 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 <math>d_a</math>, <math>d_b</math>, and <math>d_c</math> to it randomly, the resulting points will be dense in the Sierpiński triangle, so the following algorithm will again generate arbitrarily close approximations to it: Start by labelling '''p'''<sub>1</sub>, '''p'''<sub>2</sub> and '''p'''<sub>3</sub> as the corners of the Sierpiński triangle, and a random point '''v'''<sub>1</sub>. Set '''v'''<sub>n+1</sub> = ½ ( '''v'''<sub>n</sub> + '''p'''<sub>r<sub>n</sub></sub> ), where r<sub>n</sub> is a random number 1, 2 or 3. Draw the points '''v'''<sub>1</sub> to '''v'''<sub>∞</sub>. If the first point '''v'''<sub>1</sub> was a point on the Sierpiński triangle, then all the points '''v'''<sub>n</sub> lie on the Sierpiński triangle. If the first point '''v'''<sub>1</sub> to lie within the perimeter of the triangle is not a point on the Sierpiński triangle, none of the points '''v'''<sub>n</sub> will lie on the Sierpiński triangle, however they will converge on the triangle. If '''v'''<sub>1</sub> is outside the triangle, the only way '''v'''<sub>n</sub> will land on the actual triangle, is if '''v'''<sub>n</sub> is on what would be part of the triangle, if the triangle was infinitely large. [[Image:Sierpinski_chaos_animated.gif|thumb|right|200px|Animated creation of a Sierpiński triangle using the chaos game]] '''Or more simply:''' # Take 3 points in a plane, and form a triangle # Randomly select any point inside the triangle and move half the distance from that point to any of the 3 vertex points. Plot the current position. # 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 Sierpiński Gasket with a few leftover points. It is interesting to do this with pencil and paper. A brief outline is formed after placing approximately one hundred points, and detail begins to appear after a few hundred. '''Or using an Iterated function system''' An alternative way of computing the Sierpiński triangle uses an [[Iterated function system]] and starts by a point in the origin (''x''<sub>0</sub> = 0, '''y''<sub>0</sub> = 0) and then the new points are iteratively computed by randomly applying (with equal probability) one of the following three coordinate transformations (using the so called [[chaos game]]): [[Image:Sierpinski1.png|thumb|right|250px|Sierpiński triangle using IFS]] ''x''<sub>''n''+1</sub> = 0.5&nbsp;''x''<sub>''n''</sub><br> ''y''<sub>''n''+1</sub> = 0.5&nbsp;''y''<sub>''n''</sub>; a half-size copy <br> when this coordinate transformation is used, the point is drawn in yellow in the figure ''x''<sub>''n''+1</sub> = 0.5&nbsp;''x''<sub>''n''</sub>&nbsp;+&nbsp;0.5<br> ''y''<sub>''n''+1</sub> = 0.5&nbsp;''y''<sub>''n''</sub>&nbsp;+&nbsp;0.5; a half-size copy shifted right and up<br> when this coordinate transformation is used, the point is drawn in red ''x''<sub>''n''+1</sub> = 0.5&nbsp;''x''<sub>''n''</sub>&nbsp;+&nbsp;1<br> ''y''<sub>''n''+1</sub> = 0.5&nbsp;''y''<sub>''n''</sub>; a half-size copy doubled shifted to the right<br> when this coordinate transformation is used, the point is drawn in blue '''Or using an L-system''' — The Sierpiński triangle drawn using an [[L-system#Example 6: Sierpiński triangle|L-system]]. '''Other means''' — The Sierpiński triangle also appears in certain [[cellular automata]] (such as [[Rule 90]]), 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 Sierpiński triangle. ==Properties== The Sierpiński 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.{{Fact|date=August 2007}} If one takes [[Pascal's triangle]] with 2<sup>''n''</sup> rows and colors the even numbers white, and the odd numbers black, the result is an approximation to the Sierpiński triangle. More precisely, the [[limit (mathematics)|limit]] as ''n'' approaches infinity of this parity-colored 2<sup>''n''</sup>-row Pascal triangle is the Sierpiński triangle. The area of a Sierpiński 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. Therefore the proportion of even numbers in Pascal's triangle must tend to 1 as the number of rows of the triangle tends to infinity.{{Fact|date=August 2007}} ==Analogs in higher dimension== [[Image:Sierpinski pyramid.png|thumb|333px|right|A Sierpiński square-based pyramid and its 'inverse']][[Image:Sierpiński Pyramid from Above.PNG|thumb|A Sierpiński triangle-based pyramid as seen from above (5 main sections highlighted).]] <!-- This section is linked from [[Menger sponge]] --> The tetrix is the three-dimensional analog of the Sierpiński 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. This can also be done with a [[Pyramid (geometry)|pyramid]] and five copies instead.{{Fact|date=August 2007}} A tetrix constructed from an initial tetrahedron of side-length L has the property that the total surface area remains constant with each iteration. The initial surface area of the (iteration-0) tetrahedron of side-length L is <math>L^2 \sqrt{3}</math>. At the next iteration, the side-length is halved <math>L \rightarrow { L \over 2 }</math> and there are 4 such smaller tetrahedra. Therefore, the total surface area after the first iteration is: <math>4 \left( \left( {L \over 2} \right)^2 \sqrt{3} \right) = 4 { {L^2} \over 4 } \sqrt{3} = L^2 \sqrt{3}.</math> This remains the case after each iteration. Though the surface area of each subsequent tetrahedron is 1/4 that of the tetrahedron in the previous iteration, there are 4 times as many -- thus maintaining a constant total surface area. The total enclosed volume, however, is geometrically decreasing (factor of 0.5) with each iteration and asymptotically approaches 0 as the number of iterations increases. In fact, it can be shown that, while having fixed area, it has ''no'' 3-dimensional character! The [[Hausdorff dimension]] of such a construction is <math>\frac{\ln 4}{\ln 2}=2</math> which agrees with the finite area of the figure. (A Hausdorff dimension between 2 and 3 would indicate 0 volume and infinite area.) ==See also== * [[List of fractals by Hausdorff dimension]] * [[Sierpiński carpet]] * [[Triforce]] * [[Pascal's triangle]] ==References== <references/> ==External links== {{Commonscat|Sierpinski triangles}} * {{MathWorld|title=Sierpinski Sieve|urlname=SierpinskiSieve}} * [http://www.physics.utah.edu/~jasonu/code/sierpinski.cc Sierpinski Triangle C++ code] * Paul W. K. Rothemund, Nick Papadakis, and Erik Winfree, [http://biology.plosjournals.org/perlserv/?request=get-document&doi=10.1371/journal.pbio.0020424 Algorithmic Self-Assembly of DNA Sierpinski Triangles], ''PLoS Biology'', volume 2, issue 12, 2004. * [http://www.cut-the-knot.org/Curriculum/Geometry/Tremas.shtml Sierpinski Gasket by Trema Removal] at [[cut-the-knot]] * [http://www.cut-the-knot.org/triangle/Hanoi.shtml Sierpinski Gasket and Tower of Hanoi] at [[cut-the-knot]] * [http://www.phidelity.com/blog/fractal/exploring-sierpinskis-triangle-and-sacred-geometry/ Animated 3D model loop of Sierpinski's Triangle and Article on similarities with Sacred Geometry] * [http://www.stilldreamer.com/mathematics/sierpinskis_triangle/ Article explaining Sierpinski's Triangle created with a bitwise XOR (example program in Macromedia Flash ActionScript)] * [http://www.stilldreamer.com/mathematics/chaos_game/ Article explaining Sierpinski's Triangle created with the Chaos Game (example program in Macromedia Flash ActionScript)] *[http://agutie.homestead.com/files/machupicchu_sierpinski1.html Sierpinski Triangle and Machu Picchu.] Fractal illustration with animation and sound. *[http://www.visualbots.com/index.htm VisualBots] - Freeware multi-agent simulator in Microsoft Excel. Sample programs include Sierpinski Triangle. *[http://to-campos.planetaclix.pt/fractal/graftale.html IFS Fractal fern and Sierpinski triangle - JAVA applet] *[http://vispo.com/kearns Contains a section where the Sierpinski triangle can be seen step by step -- Shockwave] * The artist [[Richard Marquis]] has created [[murrine]] Sierpinski triangles which can be viewed on his website [http://www.richardmarquis.com/uploads/images/work/1670nw here], [http://www.richardmarquis.com/uploads/images/work/1741 here], and elsewhere on his [http://www.richardmarquis.com/index.php?page=recent_work recent work page]. See also the recent book by Barry Behrstock: The Way of the Artist: Reflections on Creativity and the Life, Home, Art,and Collections of Richard Marquis. *[http://www.uwosh.edu/faculty_staff/kuennene/Chaos/ChaosNotes7.pdf Another reference, removed triangles] are [[open set]]s *[http://bernsteinforpresident.com/triangle.php Online Sierpinski Triangle Generator] [[Category:Fractals]] [[Category:Factorial and binomial topics]] [[Category:Curves]] [[Category:Topological spaces]] [[Category:Triangles]] [[ca:Triangle de Sierpiński]] [[cs:Sierpinského trojúhelník]] [[de:Sierpinski-Dreieck]] [[es:Triángulo de Sierpinski]] [[eo:Triangulo de Sierpinski]] [[fr:Triangle de Sierpiński]] [[gl:Triángulo de Sierpinski]] [[ko:시에르핀스키 삼각형]] [[hr:Trokut Sierpińskog]] [[it:Triangolo di Sierpinski]] [[he:משולש סרפינסקי]] [[ja:シェルピンスキーのギャスケット]] [[pl:Trójkąt Sierpińskiego]] [[pt:Triângulo de Sierpinski]] [[ru:Треугольник Серпинского]] [[sv:Sierpinskitriangel]] [[uk:Трикутник Серпінського]] [[zh:謝爾賓斯基三角形]]