## ← 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,
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.