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The Cantor Set

The Cantor Set is a set of real numbers along the numberline from 0 to 1.

To get the set, you first take the number line from 0 to 1 and remove the middle third of it from points .333R to .666R. you now have two number lines. At the next iteration, remove the middle from the new number lines again. Do this infinitely many times to get the cantor set.

0                          .333R                      .666R                     1
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# #   # #         # #   # #                           # #   # #         # #   # #

One would think that cutting out pieces an infinite number of times would result in no numbers left at the end, but the contrary is true. There are an infinite ammount of numbers in the cantor set and they are *nowhere* dense. for example, 0, .3333R, .6666R and 1 will never be removed. neither will .25

This process of removing middle sections over and over to create smaller copies of the whole is identical to the higher dimensional analogs.

As a side note: the structure of the cantor set resembles that of a binary bifurcating tree.

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