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Functions, Graphs, and Limits

Functions, Graphs, and Limits

The Cantor Set and Fractals

Take a line segment:

Draw dots to divide the line segment into thirds:

Then erase the middle third (but keep the dots):

Repeat with the two remaining line segments. Draw dots to divide into thirds:

Then erase the middle thirds (but keep the dots):

Continue until the end of time. The limit is the Cantor Set, which will contain no line segments since any line segments have been broken up. Instead, we find infinitely many dots. We get more dots every time we break up line segments.

The Cantor Set is an example of a fractal. If we go up to 2 dimensions and do something similar, we find a collection of shapes whose limit is the Sierpinski Carpet.

If we go up yet another dimension,we discover the dizzying idea of the Menger Sponge.

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