Math 326, Feb 17 - Sierpinski's Triangle

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We build the model in 2.8, Sierpinski's Triangle.  Start by fixing the corners of an equilateral triangle at (0,0), (1,0) and (0.5,SQRT(3)/2).  Pick any starting point inside the triangle, for example (0.4, 0.1).  This starting point will be point P₀.  Now here's the plan.  We will compute the midpoint between P₀ and each of the corners of the triangle.  Pick one of these three midpoints at random, this will be P₁.  Now repeat, find the midpoint between P₁ and the corners, and pick P₂ at random from those three midpoints.  Repeat 2000 times, and graph these points.  What comes out is Sierpinski's Triangle.  Note that hitting F9 recalculates the random numbers.

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Additional Questions

• Try the same model using a square instead of a triangle.  Try to guess what the graph will look like.
• Instead of using the midpoint which is a straight average, try using a weighted average.  For example, the x coordinate of the midpoint is (0.5x₁ + 0.5x₂); instead use (ax₁ + (1-a)x₂) where "a" is a parameter (the "shrink" parameter) which we can scroll between 0 and 1.  What happens to the graph?  Make this change on the square graph as well.
• Try moving around the initial points in the triangle and the square.  For example, replace the (1,1) corner with (2,2) in the square model.  How does the graph change?  Try various corners to twist the graph in different ways.  Also try changing the shrink parameter for different shapes.