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← Intro 3.6 Box-Counting Dimension (2)

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Showing Revision 3 created 11/11/2014 by Gayle Williams.

  1. We can now experiment with box-counting dimension
  2. using the NetLogo model called
  3. boxcountingdimension.nlogo.
  4. Here you can see that this allows us to
  5. iterate examples of fractals just like we did
  6. in the previous model.
  7. So let's go for 4 iterations of this.
  8. But we can do at this point now is
  9. compare the Hausdorff dimension,
  10. 1.262 that we calculated,
  11. with a box-counting approximation.
  12. So I'm going to do Box-Counting Setup here
  13. and you can see there's an initial box length
  14. set to 10, which you can change.
  15. So here's the initial box right down here
  16. and the increment is going to be 1.0.
  17. So we're going to increase the box size
  18. by 1 unit at each iteration.
  19. Ok, so here this tells us
  20. how many boxes there are and so on.
  21. And watch over here as we do box counting
  22. where the model is going to plot
  23. the log of the number of boxes
  24. versus the log of 1 over the box length
  25. for each iteration. So let's just go ahead
  26. and do that with Box-Counting Go.
  27. Now this is just like we saw where we're
  28. putting a grid of boxes over the figure.
  29. You don't see the whole grid.
  30. You're only seeing the boxes that contain pieces of the figure.
  31. And at each time step, see iteration,
  32. we see what the box length is
  33. and the number of boxes that is being counted.
  34. And here those values are being plotted.
  35. And you see they're sort of beginning to
  36. approximate a straight line.
  37. So if we keep going, the boxes get bigger and bigger.
  38. And then we can stop it by clicking again
  39. on Box-Counting Go at any time.
  40. I haven't actually run it for very long,
  41. but I have some points and what I can do is
  42. say Find Best Fit Line.
  43. That does a linear regression
  44. and computes a box-counting dimension here of 1.122,
  45. which is a little bit different than the
  46. Hausdorff dimension of 1.262.
  47. Now that's because, remember
  48. box-counting is just an approximation.
  49. We can get a better approximation
  50. if we start with a smaller initial box length,
  51. or if we start with a smaller increment.
  52. But that, of course, is going to take longer.
  53. So let's start over with our Koch curve.
  54. Iterate, iterate, iterate...ok.
  55. And our approximation would also be improved
  56. if we iterated more.
  57. Box-counting Setup, and Go.
  58. I can speed this up, but it's still kind of a slow calculation.
  59. Net Logo is not known for its extreme speed of computing.
  60. It's kind of a trade-off.
  61. It's easy to program in, but not super fast.
  62. But anyway, now you can run this, go away.
  63. Go get a cup of coffee, like computer scientists like to do
  64. while waiting for their program to finish.
  65. And let it run for many iterations
  66. and then see how well the
  67. box-counting dimension approximates
  68. the Hausdorff dimension.
  69. And you'll see that the next exercise
  70. is to test that out.