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Line: Your job is to paint yellow centerline on the highways. Since the painted strip is only 6 inches wide, we may consider the yellow line as a theoretical line with zero width when it extends over a mile (63,360 inches) as in figure 1. The yellow paint comes in a large barrel container and it takes one barrel to paint 1-mile long yellow line. It then requires 2 barrels to paint 2-mile long yellow line, 3 barrels to paint 3-mile long yellow line, and so on, as listed in table 1.
Figure 1. Yellow centerline on highways
Table 1. Painting the yellow centerline on highways
Now, here is a helpful trick. We plot the relationship of x and y in table 1 on a special kind of graph paper called the log-log plot of figure 2. As you see, unlike regular graph paper which is linear-linear plot, both axis values for x and y are piling up toward 10 in figure 2. This is because log(1) = 0, log(2) = 0.301, log(4) = 0.602, log(8) = 0.903, and log(10) = 1, as may be checked by the 10x- key for the natural logarithmic function "log" on a pocket calculator. Figure 2 shows the plot of x versus y gives a straight line, so that we can compute its slope by rise/run of the straight line. In fact, the slope is dimension d.
The last equality in the above expression follows from log(1) = 0. Even without the use of formula (1), we see at once that d = 1 because the straight line plot is a 45° diagonal in figure 2. We have thus reconfirmed that dimension d = 1 for a line segment. Figure 2. Log-log plot of table 1 |
