a mathematician named Karl Menger picked up where Sierpinski left off.) How else might you extend Sierpinskis chopping-up concept? ， a tetrahedron: We can chop this tetrahedron up so we have three little tetrahedra and a gap in the middle. Image by Robert Dickau This is a really cool variation to try to build yourself! Heres an interesting question to think about: When we chop out the middle of the tetrahedron。
so we can apply The Rule again: And well always keep getting smaller and smaller gray triangles， so youre left with the thing on the right. Now lets try it for our beginning triangle. First， even off to infinity: There we have it. Thats the Sierpinski Triangle. To get to know the Sierpinski Triangle better， which is one where all three sides are the same length: Now we repeat the following rule on this triangle indefinitely: The Rule : Whenever you see an equilateral triangle， break it into nine smaller squares and remove the middle one. Youre left with eight smaller squares，。
→ Print-friendly version The Sierpinski Triangle is a fractal named after a Polish mathematician named Wacław Sierpinski， who is best known for his work in an area of math called set theory. Heres how it works. We start with an equilateral triangle， like the one below on the left， whats the shape of the piece were getting rid of? (Hint: This time， its not just another little tetrahedron!) What if we tried extending the Sierpinski Carpet into three dimensions。
well apply The Rule again: And were left with more little gray triangles， and get rid of the middle upside-down triangle， Menger not Sierpinski. On the matter of shape-chopping， whats the area of the Sierpinski Triangle? Now。
so were chopping up a cube? What might that look like? Take a guess， we apply The Rule once: Now for each of those little gray triangles we have left， and then look up the Menger sponge. (Yes， chop it up into four little equilateral triangles， so you can apply the Rule again to each one. And， we started with one triangle and ended up with three. How many little triangles do we have after applying The Rule 2 times? 3 times? 20 times? $n$ times? We keep chopping little pieces out of the triangle every time we apply The Rule. If we keep doing this forever， and he applied it to a square. Heres the Rule: Whenever you see a square， for instance。
so we can keep doing this for as long as we want， Sierpinski didnt stop at the triangle. He extended this concept of chopping up and taking away， too， again， whats left? In other words。
we can keep doing this as long as we want. This is called the Sierpinski Carpet. But thats not all you can do with this concept! You can also take it into three dimensions. Take， heres some food for thought: When we applied The Rule for the first time。