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336 chapter 13log N(r) = log C − d f log(r) (as r → 0), (13.19)∆N(r)⇒ d f = − lim . (13.20)r→0 ∆rC DHere s ∝ 1/r is called the scale in geography, so r → 0 corresponds to an infinitescale. To illustrate, you may be familiar with the low scale on a map being 10,000m to a centimeter, while the high scale is 100 m to a centimeter. If we want the mapto show small details (sizes), we need a map of high scale.We will use box counting to determine the dimension of a perimeter, not of anentire figure. Once we have a value for d f , we can determine a value for the lengthof the perimeter via (13.15). (If you cannot wait to see box counting in action, on theCD you will find an applet Jfracdim that goes through all the steps of box countingbefore your eyes and even plots the results.)13.5.3 Coastline Implementation and ExerciseRather than ruin our eyes using a geographic map, we use a mathematical one.Specifically, with a little imagination you will see that the top portion of Figure 13.320log(Number boxes)15105square (m = 2.00)coastline (m = 1.3)straight line (m = 1.02)0 –4 –3.5 –3 –2.5 –2 –1.5 –1log(scale)Figure 13.5 Dimensions of a line, box, and coastline determined by box counting.−101COPYRIGHT 2008, PRINCET O N UNIVE R S I T Y P R E S SEVALUATION COPY ONLY. NOT FOR USE IN COURSES.ALLpup_06.04 — 2008/2/15 — Page 336