21
Oct
09

Sierpinski’s Triangle


In the last post on fractals, I gave examples to describe the concept of fractals. Fractals are figures with fractional dimesions.

One of the simplest of them is the Sierpinski’s Triangle.

Sierpinskis Triangle

Sierpinski's Triangle

Constructing a triangle

  1. Start with any triangle in a plane (any closed, bounded region in the plane will actually work). The canonical Sierpinski triangle uses an equilateral triangle with a base parallel to the horizontal axis
  2. Shrink the triangle to ½ height and ½ width, make three copies, and position the three shrunken triangles so that each triangle touches the two other triangles at a corner . Note the emergence of the central hole – because the three shrunken triangles can between them cover only 3/4 of the area of the original. (Holes are an important feature of Sierpinski’s triangle.)
  3. Repeat step 2 with each of the smaller triangles.

The Sierpinski triangle has 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. (and hence a fractal)

Leaving you with an applet on Sierpinski’s triangle. Try it and you can see the triangle build step by step.

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