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| Left: Saturn's rings (NASA). Right: a product of a Cantor set and a circle. |
| |
| J. Avron and B. Simon showed that the apparent fractality of the
ring structure could be caused by gravitational resonances. |
| However, what they found was not a familiar Cantor set, such as the
middle-thirds set, but rather a fat Cantor set. |
| The middle-thirds set is a thin Cantor set, as are
all the others we've seen so far. Here the term "thin" means the
Cantor set has length 0. To see why this is true, recall the middle-thirds
set is constructed from the unit interval by removing one interval
of length 1/3, two intervals of length 1/9,
four intervals of length 1/27, and so on. The lengths of the
removed intervals add up to |
| 1/3 + 2/9 + 4/27 + ... | = (1/3)(1 + 2/3 + 4/9 + ...) |
| = (1/3)(1/(1 - 2/3))) = 1 |
|
| where we have used the sum of the geometric series |
| 1 + r + r2 + r2 + ... = 1/(1 - r) |
| so long as |r| < 1. Because the lengths of the intervals removed sum to the
length of the starting interval, the middle-thirds set has length 0. It is thin. |
| To build a fat Cantor set, just multiply the lengths of the removed intervals by
the same number, 1/2 for example. That is, we remove
one interval of length 1/6, two of length 1/18, 4 of length
1/54, and so on. The removed lengths add up to 1/2, so the Cantor set
formed this way has length 1/2. It is fat. |
| (Fat Cantor sets aren't self-similar, because the scaling factor varies with
each additional level of intervals removed.) |
| In fact, this fat Cantor set has dimension 1, so the product of this fat
Cantor set and a circle has dimension 2. |
| If the dimension of Saturn's rings were less than 2, the rings would
reflect almost no light and so would be almost invisible. If Avron and Simon
had gotten a thin Cantor set for the ring cross-section, we'd know their
derivation was wrong. But a fat Cantor set is consistent with what we observe. |