Sierpinski triangle

File:Sierpinski triangle.png
Sierpinski triangle

The Sierpinski triangle, also called the Sierpinski gasket, is a fractal, named after Wacław Sierpiński. Sierpiński demonstrated that this fractal is a universal curve, in that any possible one-dimensional graph, projected onto the two-dimensional plane, is homeomorphic to a subset of the Sierpinski gasket. For curves that cannot be drawn on a 2D surface without self-intersections, the corresponding universal curve is the Menger sponge.

Construction

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

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 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 $\displaystyle d_a$ 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 $\displaystyle d_a$ U $\displaystyle d_b$ U $\displaystyle d_c$ .

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 $\displaystyle d_a$ , $\displaystyle d_b$ , and $\displaystyle d_c$ 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 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

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 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.

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.

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).