A fractal, also known as the Sierpinski
Triangle or Sierpinski Sieve after its inventor
Waclaw Sierpinski. It is produced by the following
set of rules:
The gasket can also be made by starting with Pascal's
Triangle, then coloring the even numbers white and the odd numbers black.
Most curiously, it can be generated by a game of chance. Start with three
points, labeled 1, 2, and 3, and any starting point, S. Then select
randomly 1, 2, or 3, using a die or some other method. Each random number
defines a new point halfway between the latest point and the labeled point
that the random number indicates. When the game has gone on long enough
the pattern produced is the Sierpinski Gasket.
- Start with any triangle in a plane.
- Shrink the triangle by ½, make three copies, and translate them
so that each triangle touches the two other triangles at a corner.
- Repeat step 2 ad infinitum.
The Gasket has a Hausdorff dimension
of 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 ½. Adding rounded
corners to the defining curve gives a non-intersecting curve that traverses
the gasket from one corner to another and which Benoit Mandelbrot
called the Sierpinski Arrowhead.
AND PATHOLOGICAL CURVES