HandsOn 6 - 1-, 2-, & 3-Dimensions
Fill in a copy of Table 2.4. Now plot the entries in this table on a fresh copy of the log-log graph paper. Along the horizontal axis plot the number of segments and, on the vertical axis, the entries in columns 2, 3, and 4. Draw a line through each set of points and verify that a line, square, and cube are 1-, 2-, and 3-dimensional, respectively.
l | Number of segments | Number of squares | Number of cubes |
of length d needed | of side d needed | of edge d needed | |
to span line | to cover square | to fill cube | |
of length L = 100 | of area L^{2} = 10,000 | of volume L^{3} = 10^{6} | |
1 | |||
2 | |||
5 | |||
10 | |||
20 | |||
50 | |||
100 |
Note that in Figure 2.6 the dimension is the exponent in the equation for length or area or volume, as a function of L.
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