Ron Menendez of Chatham, N.J., has noted yet another interesting property of the Sierpinski gasket (be/ow),which I discussed in "Sierpin-ski's Ubiquitous Gasket" [August 1999]. Take an equilateral triangle with vertices A, B and C and pick any point X in the triangle's plane. Choose one vertex at random—for example, roll a die and let 1 or 2 correspond to A,3 or 4 to B,and 5 or 6 to C.Find the midpoint of the line joining X to the chosen vertex:this is the new position of X. Now repeat the procedure, always choosing a random vertex and moving X to the midpoint between its previous position and that vertex. Aside from a few initial points where the random walk is "settling down," the resulting cloud of points is a Sierpinski gasket!

This surprising outcome is explained by mathematician Michael Barnsley's theory of self-similar fractals. The Sierpinski gasket has three corners, which can also be labeled A, B and C.It is made from three smaller copies of itself,each with sides half as long as the gasket's side.If you draw a line between any point in the gasket and A,B or C,the midpoint of the line will also lie in the gasket. This feature corresponds to A^A the rules for Menendez's A random walk.Barnsley has proved that any such walk "converges'

0 0

Post a comment