Sierpinski Carpet

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Investigate the iteration of

$x \rightarrow 1-x $ with different starting points.

The Sierpinski Carpet

To construct this set, start with a filled-in square. Divide this square into nine squares of equal size and remove the centre one as shown. Repeat this procedure with each of the eight remaining squares, and continue in the same manner.

The Sierpinski Carpet.

 

The first stage of the gasket is a square of side length 1 unit.

What is the total length of the boundary of the set left in at the nth stage?

What is the total area of all squares taken out up to and including the nth stage?

What is the total area of that part left in at the nth stage?

What happens as n gets larger and larger?

 

Discuss the self-similar nature of the gasket.

What happens if we start with a parallelogram instead of a square?

 

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