We wrapped up Thursday’s lesson with an explanation of the idea of the Hausdorff Dimension of a fractal. In brief, the Hausdorff Dimension is the solution to the equation S^d = N, where S is the scale by which a fractal is being broken up into pieces, and N is the number of such pieces. So for the Sierpinski Triangle, we cut the original triangle into pieces that are all half the length of the original (S = 2), but keep 3 of those pieces. Solving the equation 2^d = 3 gives d ≈ 1.585, which is the Hausdorff dimension of the Sierpinski Triangle. For the Koch Curve, we get the equation 3^d = 4, producing d ≈ 1.26, and for the Cantor Set, we get 3^d = 2, so d ≈ 0.631.
For homework, you’ve been asked to think about the dimension of the Dragon Curve, and have been reminded that the base template for the curve involves a right isosceles triangle, as well as to design a fractal called the Sierpinski Carpet (essentially the same design as the Sierpinski Triangle, but with a square as the starting shape) and to find the dimension of that.