Sierpinski triangle

The Sierpinski triangle, also called the Sierpinski gasket, is a fractal, named after Waclaw Sierpinski.

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

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Sierpinsky_triangle_(evolution).gif
The evolution of the Sierpinsky triangle

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 1/2, make three copies, and position the three copies 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 Sierpinsky gasket. Michael Barnsley was using an image of a fish to illustrate this in his paper V-variable fractals and superfractals (http://wwwmaths.anu.edu.au/~barnsley/pdfs/V-var_super_fractals.pdf) (PDF).

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Sierpinski_triangle.png
Sierpinski triangle
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Fractal_squares.png
Sierpinski carpet

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 [itex]d_a[itex] note the dilation by a factor of 1/2 about a point a, then the Sierpinski triangle with corners a, b, and c is the fixed set of the transformation [itex]d_a[itex] U [itex]d_b[itex] U [itex]d_c[itex].

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 [itex]d_a[itex], [itex]d_b[itex], and [itex]d_c[itex] 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 vn+1 = ½ ( vn + prn ), where rn is a random number 1, 2 or 3. Draw the points v1 to v. If the original point vn was a point on the Sierpinski triangle, then all the points vn lie on the Sierpinski triangle. If the first point vn to lie within the perimeter of the triangle is not a point on the Sierpinski triangle, none of the points vn will lie on the Sierpinski triangle, however they will converge on the triangle. If v1 is outside the triangle, the only way vn will land on the actual triangle, is if vn 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 of the three points and move half the distance to there. Plot the current position.
3. Repeat from step 2

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.

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.

The black area of a Sierpinski triangle is zero. This can been seen from the infinite iteration, where we remove 25% of the area left at the previous iteration.

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