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← Intro 3.6 Box-Counting Dimension (1)

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Showing Revision 5 created 11/11/2014 by Gayle Williams.

  1. In the previous subunit I talked about
  2. the fractal dimension of various objects
  3. such as coastlines.
  4. But I haven't yet told you how these
  5. real world fractal dimensions were computed.
  6. It was possible for us to compute
  7. fractal dimension from the Koch curve
  8. and the Sierpinski triangle because
  9. these are perfect mathematical fractals,
  10. not real world objects.
  11. But there's a lot of interest in computing
  12. approximate fractal dimension in the real world
  13. because it can often reveal insights about
  14. natural or human created systems.
  15. There are a lot of different methods for analyzing
  16. fractals and whole books devoted to this subject.
  17. Here I'm going to show you one commonly used
  18. method for estimating fractal dimension,
  19. the box counting method.
  20. The box counting method is
  21. closely related to this idea that
  22. as you change the size of the ruler
  23. that you measure a fractal by,
  24. you get a different length as you go
  25. further and further into smaller and smaller length scales.
  26. So here's what the box counting method consists of.
  27. You take a particular object.
  28. Here I have a picture of the British coastline.
  29. So what we do is overlay this figure
  30. by a grid of boxes.
  31. Each box has a certain length of its side,
  32. which is the scale at which we're measuring this figure.
  33. And what we do is count
  34. the number of boxes in which
  35. part of the black outline of the coast appears.
  36. For example, it does not appear in this box,
  37. even though this is in the middle of Great Britain,
  38. so we don't count it.
  39. So if we follow that procedure
  40. and count the number of boxes
  41. containing part of this black outline,
  42. I got 36.
  43. The length of the side was 10 units for each box.
  44. Now I go to the next step
  45. and I increase the size of the boxes.
  46. So I'm now calculating the number of boxes,
  47. but at a different scale.
  48. Here because the length of the side
  49. of the box was larger, I got fewer boxes
  50. that contained part of this figure.
  51. Then I would go up again.
  52. Here the size of the box is larger again, 12.
  53. And I got 27 boxes that contained part of the figure.
  54. So you keep doing this,
  55. accumulating this list of numbers.
  56. Let's look at the relationship
  57. between Hausdorff dimension,
  58. which we already learned about,
  59. and box-counting dimension.
  60. If you recall, for the Hausdorff dimension
  61. we had a relationship that is
  62. the number of copies of a figure
  63. from a previous level.
  64. If we take the log of that,
  65. that was equal to the dimension
  66. times the log of the reduction factor
  67. from the previous level.
  68. It can be shown that if you do this
  69. box-counting method, this can be approximated
  70. by looking at the log of the number of boxes
  71. and that's equal to the dimension
  72. times the log of 1 over the length of the side.
  73. D is called the box-counting dimension
  74. and if you want to see the derivation of this
  75. and other details about the relationship
  76. between these dimensions,
  77. take a look at Chapter 4 of the Fractal Explorer
  78. which is a website about fractals.
  79. And there's a link from our
  80. Course Materials page on this.
  81. Now the question is,
  82. how do we actually get this D from our values
  83. from numbers of boxes and
  84. lengths of sides.
  85. Well if you're up on your algebra
  86. you might have noticed that this equation
  87. is actually the equation of a straight line.
  88. If we plot it on a graph
  89. where the axes are here,
  90. the log of one over the length of the side,
  91. this x value,
  92. and the y axis is log of the number of boxes.
  93. And D would be the slope of that straight line.
  94. So what we can do is we can
  95. take the measurements that we made
  96. at each level for the box counting
  97. and we can plot it, each measurement, on this graph.
  98. So here's some hypothetical measurements
  99. that we might have gotten,
  100. where the number of boxes goes down
  101. as the length of the side goes up.
  102. Notice this is 1 over the length of the side,
  103. so as length of the side goes up, this goes down.
  104. You can see that if this is actually true
  105. these should form a straight line
  106. whose slope is the dimension.
  107. So we can estimate the dimension
  108. by plotting these points,
  109. doing our measurements for the boxes
  110. and then plotting these points.
  111. Drawing a straight line through them,
  112. figuring out what the slope of that line is,
  113. and that's our measured dimension.
  114. And that's roughly what people did
  115. to calculate things like the
  116. fractal dimension of coastlines.