Hello, TreeMoney!

Good ol' Cantor and Sierpinski . . . with mind-boggling concepts.

. .

. .

There is 1 segment of length

. . . . . . 2 segments of length

. . . . . . 4 segments of length

. . . . . . 8 segments of length . . . and so on.

The total length is: .

Hence: .

We have a Geometric Series with first term and common ratio

Its sum is: .

Evidently, *all* of the interval has been removed.

Yet there are infinitely many numbers remaining in the Cantor set.

I'll let you figure out which ones are left.

b) The Sierpinski carpet is a two-dimensional counterpart of the Cantor set.

It is constructed by removing the center one-ninth of a square of side 1,

then removing the centers of the eight smaller remaining squares, and so on.

Show that the sum of the areas of the removed squares is 1.

This implies that the Sierpinski carpet has area 0. Code:

*-----------------------------------*
| ϧ ϧ ϧ ϧ ϧ ϧ ϧ ϧ ϧ |
| *---* *---* *---* |
| ϧ |:::| ϧ ϧ |:::| ϧ ϧ |:::| ϧ |
| *---* *---* *---* |
| ϧ ϧ ϧ ϧ ϧ ϧ ϧ ϧ ϧ |
| *-----------* |
| ϧ ϧ ϧ |:::::::::::| ϧ ϧ ϧ |
| *---* |:::::::::::| *---* |
| ϧ |:::| ϧ |:::::::::::| ϧ |:::| ϧ |
| *---* |:::::::::::| *---* |
| ϧ ϧ ϧ |:::::::::::| ϧ ϧ ϧ |
| *-----------* |
| ϧ ϧ ϧ ϧ ϧ ϧ ϧ ϧ ϧ |
| *---* *---* *---* |
| ϧ |:::| ϧ ϧ |:::| ϧ ϧ |:::| ϧ |
| *---* *---* *---* |
| ϧ ϧ ϧ ϧ ϧ ϧ ϧ ϧ ϧ |
*-----------------------------------*

There is: 1 square with area

. . . . . . . 8 squares with area

. . . . . . . 64 squares with area

. . . . . . . 512 squares with area . . . and so on.

The total area is: .

Hence: .

We have a Geometric Series with first term and common ratio

Its sum is: .