THE DIMENSION OF THE MASS COAST LINE Math 21b, O.KnillTo measure the dimension of an object, one can count the number f (n) of boxes of length 1/n needed to coverthe object and see how f(n) grows. If f (n) grows like n2, then the dimension is 2, if n(k) grows like n, thedimension is 1. For fractal objects, like coast lines, the number of boxes grows like nsfor a number s between1 and 2. The dimension is obtained by correlating yk= log2f(k) with xk= log2(k). We measure:The Massachusetts coast line is a fractal of dimension 1.3.We measure the data: f(1) = 5, f(2) = 12,f(4) = 32 and f(8) = 72. A plot of the data(xk, yk) = (log2(k), log2f(k)) together with aleast square fit can be seen to the right. Theslope of the line is the dimension of the coast.It is about 1.295. In order to measure the di-mension better, one would need better maps.xkyk0 log2(5)1 log2(12)2 log2(32)3 log2(72)Finding the best linear fit y = ax + b is equivalent to find the least square solution of the system0a + b = log2(5), 1a + b = log2(12), 2a + b = log2(32), 3a + b = log2(72)which is A~x =~b with ~x =ab,~b =2.33.556and A =0 11 12 13 1. We have ATA =14 66 4, (ATA)−1=2 −3−3 7/10 and B = (ATA)−1AT=−3 −1 1 37 4 1 −2/10. We get =ab= B~b =1.292.32.COMPARISON: THE DIMENSION OF THE CIRCLELet us compare this with a smoothcurve. To cover a circle withsquares of size 2π/2n, we needabout f(n) = 2nto cover the cir-cle. We measure that the circle isnot a fractal. It has dimension 1.xkyk3 log2(8) = 34 log2(16) = 45 log2(32) = 56 log2(64) = 6COMPARISON: THE DIMENSION OF THE DISKFor an other comparison, we takea disk. To cover the disk withsquares of size 2π/2n, we needabout f(n) = 22nsquares. Wemeasure that the disk is not a frac-tal. It has dimension 2.xkyk3 log2(5) = 2.34 log2(13) = 3.75 log2(49) = 5.616 log2(213) = 7.73REMARKSCalculating the dimension of coast lines is a classical illustration of ”fractal theory”. The coast of Brittain hasbeen measured to have a dimension of 1.3 too. For natural objects, it is typical that measuring the length ona smaller scale gives larger results: one can measure empirically the dimension of mountains, clouds, plants,snowflakes, the lung etc. The dimension is an indicator how rough a curve or surface
View Full Document
Unlocking...