I know that the fraction of shaded area reduces at each stage by 3/4 but my question is that is: Is there a fraction of shaded area that must remain regardless of the number of stages? If there is one, what is the fraction :O?
Thanks alot ! :D
Thanks alot ! :D
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Let the area of the large white triangle be 1
To make the first black triangle that will be glued on the large white triangle (the one that connects the midpoints of its sides), a black triangle the same size as the white one is folded in a way that actually makes 4 smaller black triangles. So the area of the first black triangle is 1/4.
A1= 1/4
A black triangle whose size is 1/4 will be cut into 4 smaller triangles. Each of these triangles has an area of (1/4) X (1/4) = 1/16. Three of them will be placed inside the white triangles. Then the total black area will be:
A2= 1/4+3*1/4*1/4= 1/4 + 3(1/4²)
The next triangles to place on the Sierpinski Gasket will be a quarter the size of the 1/16 triangles, or 1/64, and there will be 3 X 3 = 9 of them, making the total area:
A3= 1/4 + 3(1/4²)+ 3²(1/4^3)
There will be 9 X 3 = 27 of the next size smaller triangles. Since they are a quarter of the size of the previous ones, their size will be 1/256. When those triangles are places, the total area covered by black triangles is:
A4= = 1/4 + 3(1/4²)+ 3²(1/4^3)+ 3^3(1/4^4)
so that fraction is 1/4
To make the first black triangle that will be glued on the large white triangle (the one that connects the midpoints of its sides), a black triangle the same size as the white one is folded in a way that actually makes 4 smaller black triangles. So the area of the first black triangle is 1/4.
A1= 1/4
A black triangle whose size is 1/4 will be cut into 4 smaller triangles. Each of these triangles has an area of (1/4) X (1/4) = 1/16. Three of them will be placed inside the white triangles. Then the total black area will be:
A2= 1/4+3*1/4*1/4= 1/4 + 3(1/4²)
The next triangles to place on the Sierpinski Gasket will be a quarter the size of the 1/16 triangles, or 1/64, and there will be 3 X 3 = 9 of them, making the total area:
A3= 1/4 + 3(1/4²)+ 3²(1/4^3)
There will be 9 X 3 = 27 of the next size smaller triangles. Since they are a quarter of the size of the previous ones, their size will be 1/256. When those triangles are places, the total area covered by black triangles is:
A4= = 1/4 + 3(1/4²)+ 3²(1/4^3)+ 3^3(1/4^4)
so that fraction is 1/4