Pythagoras twisted squares: Why did they not teach you any of this in school?

infinite

squares

, on their journey the ideal insects will circle the center infinitely times. That's a little unexpected, isn't it? Now, does

this

mean that the path of an error is infinitely long? Not really, although these paths rotate infinitely around the center,

they

have a finite length. Let's find out how long one of these paths is. If the side of the square is 1 unit long, an insect walks 1/5 of 1 and 1/5 of 1 is 0.2. The remaining distance is 0.8. We find the length of the side of the smallest square with Pythagoras. It is approximately 0.825. Since the length of outer side 1 is reduced to 0.825 in the second square, we conclude that 0.825 is the factor by which the

squares

are reduced.

This allows us to calculate the lengths of all the segments of one of the error paths. Good? Since the first segment is 0.2 long, the second segment is 0.2 times 0.825 long. To get the length of the next segment we just have to multiply by 0.825 again and so on. This means that the total length of the path is

this

. The infinite series in parentheses is a geometric series. Since you're still here, you probably know the formula for the sum of a geometric series by heart. It's just 1 out of 1 minus the scale factor. With this, the total length of a path turns out to be the following: this is approximately 1.14, which is a little longer than one of the sides of the square.

Interesting. If we reduce the fraction of sides covered by insects from 0.2 to 0.1 the numbers change like this. If we reduce the fraction of sides covered by insects from 0.2 to 0.1 the numbers change like this. Aha, the path length is even closer to 1. In fact, if we let the fraction of sides covered by the insects reach 0, the image will become the image corresponding to Martin Gardner's original problem. And you can see that the length of the insect paths in the original problem is exactly 1. What a good answer that is :) By the way, this was the cover image of Martin Gardner's Scientific American article.

Again, the insect paths are exactly as long as the sides of the square. It's a pretty surprising answer, yes, but in retrospect, maybe there's a simpler way to see why the insects' route should be exactly as long as their original distance. Again leave your opinion in the comments. I could report on many more ingenious results when it comes to finding bugs, but let's leave it for today. If you're interested in more interesting details on any of the topics I mentioned today, check out the links in the video description. One last interesting idea. At first we encounter two types of

twisted

square diagrams.

This one and that one. Actually there is a third. In this third diagram the triangles overlap. Here then is the final puzzle. Let's say A=1/2 and B=1 as in thisFor example, what is the area of ​​the square that is in the middle? Once again, leave your hopefully surprising answers in the comments. Okay and that's all from me for today. As promised, I leave you with an animation of the proof of the Cauchy-Schwarz inequality in the case of two numbers A and B.