Question

Hello,

Here is a question with something i've never heard of.

These are the first two stages of a fractal known as the Sierpinski carpet. The carpet begins with a square (stage 1). The square is cut into 9 congruent squares and the middle square is removed (stage 2).

Draw stage 3 using the method described to get from stage 1 to stage 2.

To the nearest whole percent, what percent of the original square remains in stage 3? Show your work and explain your answer.

Answer 1

ok, I realize that I should cut each remaining square into 9 quadrants leaving the middle open of each of them.

Now, I just don't know how to get the percent of the original square that remains in stage 3.

Answer 2

In stage one say the square is 9 units by 9 units thus 81 square units in area.

Then we cut it into 9 units each 3 units by three units so each 9 square units in area. We remove one so we only have 8 left each 9 square units in area.

Now for stage 3 we split each of those 8 squares into 9 squares each one unit by one unit or one unit square. We now divide one of those 1 by 1 squares into 9 squares each 1/3 by 1/3 so we can remove the middle one. The area left is 1 minus the middle which is 1/3*1/3 = 1/9 so it is 8/9

so we started with a square of 9 by 9 or 81 square units and now we have a little square with area of 8/9 square units so
(8/9)/81 is the ratio of area of one of the final little carpets to the original carpet.

Answer 3

Remember you have 8 of those little carpets.

Answer 4

This might help, nice drawing of it.
(Broken Link Removed)