Sierpinski's Triangle - 1
Sierpinski's Triangle is a self-similar fractal. To construct it do the following:
Take a red equilateral triangle of area 1. For each red triangle:
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Find the midpoint of all sides.
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Connect the midpoints.
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Shade the formed triangle white.
If you repeat these steps (in the bullet point list) an infinite amount of times, the limit will be Sierpinski's triangle.
Let step 0 be the step for which the triangle is completely red.
Colors are chosen at random.
Question: In step of Sierpinski's triangle, there are red triangles, how many red triangles will there be in step ?
See the next problem .
Note: This problem is part of a set of problems, further investigating Sierpinski's Triangle in different ways. If you want to continue investigating click here . Questions will increase in difficulty.
Image Credits: Sierpinski's Triangle Image created by William Andrus. Removed grey outline.
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1 solution
At step 0 we have only 1 red triangle
At step 1 the previous red triangle is divided into four identical triangles, 3 of which are red . Hence, there are 3 ( 1 ) = 3 red triangles .
At step 2 each red triangle in the previous step is divided into four identical triangles, 3 of which are red. Therefore, since there were 3 red triangles in step 1 , now there will be 3 ( 3 ) = 9 red triangles .
At step 3 each red triangle in the previous step is divided into four identical triangles, 3 of which are red. Therefore, since there were 9 red triangles in step 2 , now there will be 3 ( 9 ) = 2 7 red triangles .
Repeating this process, we get that if at step n there x red triangles then at step n + 1 we will have 3 x of them.