Fractal dimensionThe formula (1) has worked out correctly for a line, square, and cube to recover the Euclidean dimension 1, 2, and 3, respectively. So, we now proceed to compute dimension of the fractal objects introduced in Lesson 5. Cantor set: We again consider the scenario of painting the yellow centerline but on a highway that is infrequently traveled. The highway maintenance department has noticed that, although the traffic is busy within a 1-mile radius of towns, the highways in between towns that are farther apart are traveled lightly most of the time. Specifically, the longer the distance between towns, the less the traffic measured in terms of the number of two cars passing each other in opposite directions. So, the following Cantor painting strategy is proposed: (1) Paint a continuous yellow centerline when the towns are 1 mile apart. (2) We leave out the yellow centerline in the middle 1-mile stretch when the towns are 3 miles apart. (3) However, for the 9-mile highway we leave out the yellow centerline in the middle 3-mile section and then use the 3-mile painting strategy (2) for the highways next to the town. This is illustrated in figure 5. Again, assuming one barrel of yellow paint for a mile of centerline strip, table 4 presents paint requirement for the Cantor painting strategy.
Figure 5. The Cantor painting strategy Table 4. Paint requirement for the Cantor painting
Figure 6. Log-log plot of table 4The plot of x versus y gives a straight line in figure 6, but the slope is less than one because it lies below the diagonal (see, figure 2). By formula (1) we find that
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0.631, which is not an integer dimension. A fractal object can have either an integer or non-integer dimension, as we shall see later. Here, intuitively, that dimension is less than one makes sense because the Cantor painting requires less paint than the solid centerline painting. Hence, the fractal dimension of d 
0.631 is a measure of brokenness in the yellow centerline.