SIERPINSKI TRIANGLE

The 'Sierpinski triangle', also called the 'Sierpinski gasket', is a fractal named after Wacław Sierpiński who described it in 1915.^[1]
Originally constructed as a curve, this is one of the basic examples of self-similar sets, i.e. it is a mathematically generated pattern that can be reproducible at any magnification or reduction.

Contents

Construction

Properties

Analogs in higher dimension

See also

References

External links

Construction

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

#Start with any triangle in a plane (any closed, bounded region in the plane will actually work). The canonical Sierpinski triangle uses an equilateral triangle with a base parallel to the horizontal axis (first image).
#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).
#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).

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 'p'₁, 'p'₂ and 'p'₃ as the corners of the Sierpinski triangle, and a random point 'v'₁. Set 'v'_n+1 = ½ ( 'v'_n + 'p'_rn ), where r_n is a random number 1, 2 or 3. Draw the points 'v'₁ to 'v'_∞. If the first point 'v'₁ was a point on the Sierpinski triangle, then all the points 'v'_n lie on the Sierpinski triangle. If the first point 'v'₁ 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 'v'₁ 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 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 Sierpinski 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 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):

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
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 .
'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.

Properties

The Sierpinski triangle has Hausdorff dimension ≈1.585, which follows from the fact that it is a union of three copies of itself, each scaled by a factor of .
If one takes Pascal's triangle with 2^''n'' rows and colors the even numbers white, and the odd numbers black, the result is an approximation to the Sierpinski triangle. More precisely, the limit as ''n'' approaches infinity of this parity-colored 2^''n''-row Pascal triangle is the Sierpinski 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. 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.

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. This can also be done with a pyramid and five copies instead.

See also

★ List of fractals by Hausdorff dimension

★ Sierpinski carpet

★ Triforce

References

1. . W. Sierpinski,
''Sur une courbe dont tout point est un point de ramification'',
C. R. Acad. Sci. Paris 160(1915) 302-305

External links

★

★ Sierpinski Triangle C++ code

★ Paul W. K. Rothemund, Nick Papadakis, and Erik Winfree, Algorithmic Self-Assembly of DNA Sierpinski Triangles, ''PLoS Biology'', volume 2, issue 12, 2004.

★ 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.

★ IFS Fractal fern and Sierpinski triangle - JAVA applet

★ Contains a section where the Sierpinski triangle can be seen step by step -- Shockwave