Posted by JOE on Wednesday, February 10, 2010 at 8:54pm.
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.

math  JOE, Wednesday, February 10, 2010 at 9:17pm
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.

math  Damon, Wednesday, February 10, 2010 at 9:17pm
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.

math  Damon, Wednesday, February 10, 2010 at 9:20pm
Remember you have 8 of those little carpets.

math  Damon, Wednesday, February 10, 2010 at 9:25pm
This might help, nice drawing of it.
(Broken Link Removed)
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