corners A, B, and C is the fixed set of the transformation dA ∪ dB ∪ dC. 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. Chaos game If one takes a point and applies each of the transformations dA, dB, and dC 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 labeling p1, p2 and p3 as the corners of the Sierpinski triangle, and a random point v1. Set , where rn is a random number 1, 2 or 3. Draw the points v1 to v∞. If the first point v1 was a point on the Sierpiński triangle, then all the points vn lie on the Sierpiński triangle. If the first point v1 to lie within the perimeter of the triangle is not a point on the Sierpiński triangle, none of the points vn will lie on the Sierpiński 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 more simply: Take three points in a plane to form a triangle. Randomly select any point inside the triangle and consider that your current position. Randomly select any one of the three vertex points. Move half the distance from your current position to the selected vertex. Plot the current position. Repeat from step 3. This method is also called the chaos game, and is an example of an iterated function system. You can start from any point outside or inside the triangle, and it would eventually form the Sierpiński Gasket with a few leftover points (if the starting point lies on the outline of the triangle, there are no leftover points). With pencil and paper, a brief outline is formed after placing approximately one hundred points, and detail begins to appear after a few hundred. Arrowhead construction of Sierpiński gasket Another construction for the Sierpinski gasket shows that it can be constructed as a curve in the plane. It is formed by a process of repeated modification of