Mathematics is a strange and mysterious thing. It is as if it is almost magical, able to do things that our physical beings cannot percieve. Perhaps a very good example of this is a problem that I encountered just yesterday, having to do with the Sierpinski Triangle.
First of all, you might ask, what in the world is a Sierpinski triangle? Well, it is a fractal, in which it creates a self-similar pattern within itself. Perhaps a picture is better-
As you can see, the pattern here is just forming triangles inside every dark triangle on and on and on. In fact, this pattern can go on for infinity, as it will in the problem I will be showcasing below:
The segments joining the midpoints of the sides of an equilateral triangle are drawn, and the interior of the triangle they form is removed from the interior of the original triangle. The segments connecting the midpoint of the remaining triangles are joined, and the interiors of the triangles they form are removed from the interior of the original triangle. If this process is repeated forever, how much of the orginial interior will be left?
Let’s take a look back at the picture I showed you above. Suppose the black spaces were the interior of the triangle. If we were to remove the triangles, they would then leave white triangles behind. In other words, the black spaces are the interior, and the white spaces are the places where triangles have been removed. Now let’s go back to the problem. What are we trying to find? Well, we are trying to find out how much of the original interior (aka how much of the first triangle in the picture above) is left. So how do we approach this?
Perhaps we can find the total amount of area that was removed from the original triangle, or in other words, the total area of white space. We would then get that and subtract it from the area of the original triangle. This difference (aka the amount of black space left after all the removing) would then be divided by the area of the original triangle to get a fraction of how much is left of the original triangle. That’s our process.
Let’s begin. First thing- how do we calculate the total amount of space removed? In other words, how do we calculate the total amount of white space? Well, let’s take this as an infinite geometric sequence. In our first cut, ¼x is removed, where x is the original triangle’s area. This is indicated by the second triangle in the picture above. In our second cut, 3 (1/16) x is removed, as indicated by the third triangle above. In our third cut, 9 (1/64) x is removed, indicated by the fourth triangle. This pattern then continues on infinitely. Again, it is a sequence- (1/4)x, 3(1/16)x, 9(1/64)x,…… So in order to find the total amount of white space, we have to add all the terms here up. You might ask, well this is infinite, so how in the world do we add this up? Note that the first term is (1/4)x and the ratio is 3/4. We now use the formula for an infinite geometric series:
where a1 is the first term, and r is the common ratio. Both of these we know. So let’s substitute. S= [(1/4)x] / [(1- 3/4)] = [(1/4)x] / [1/4] = x. Thus, a total of x is removed from the triangle.
But wait, isn’t x the area of the original triangle, too? This means that if a total of x is removed, then in essence, the whole triangle has disappeared. Thus, the answer is that zero percent of the original interior is left. Everything has disappeared. Now picture this in your head. You are given a paper triangle. You are to cut out a triangle whose edges are the midpoints of the original triangle. You do this for every triangle that you see after you cut every time. And not only that, this is repeated infinitely. If so, then in the end, you get nothing.
To me, that’s really hard to imagine. How in the world do you get nothing if you keep on cutting a paper? It’s crazy and bizarre. Yet, at the same time, this is what makes math beautiful. And in this case, seemingly magical.