3 ms·

This article (linked elsewhere on this page) is a good description: http://www.irregularwebcomic.net/2339.html http://www.irregularwebcomic.net/2339.html Once

This article (linked elsewhere on this page) is a good description: http://www.irregularwebcomic.net/2339.html http://www.irregularwebcomic.net/2339.html Once I understood this, I find it is a bit a cheat (as you might expect). The 'cuts' are not cuts in any real-world sense. Hopefully you are happy with the idea that there are as many integers as there are both even integers or odd integers. Therefore in some sense I can "cut" the integers into two equally sized sets. This paradox does something very similar on the real numbers, making use of the form of the real numbers to avoid the "doubling" effect I got when cutting the integers.

ColinWright 14y ago
It doesn't do it on the real numbers at all, it does it on a subset of rotations in R^3. You're right that these aren't "pieces" in the intuitive visual sense. That's why Feynman ridiculed the result, because in the initial description given to him he assumed physical pieces, not pairwise disjoint sets. http://news.ycombinator.com/item?id=411307 http://news.ycombinator.com/item?id=411307 There are similar paradoxical dissections of the plane: http://news.ycombinator.com/item?id=411637 http://news.ycombinator.com/item?id=411637
- CJefferson 14y ago
  You are right, I was oversimplifying perhaps too far!
  - ColinWright 14y ago
    But it is a way in to understanding the way things work. It helps to have simpler situations that exhibit aspects of the whole thing. I remember being told about the library that has all finite books that can possibly be written. Every finite string of characters appears as a book. Obviously there are infinitely many, but we won't worry too much about that. Now we find that someone, overnight, stole all the books except for those that start with the letter "q". We're annoyed at first, but then someone suggests we just erase the initial "q" from all the remaining books. So we do that, and lo and behold! We have the full collection again! We even now have an extra book, one that's completely blank! It's not a paradox, but it is the way infinite things can work, and it shows how 1/26 of a collection can be the same as the whole collection. This is related to how the B-T "paradox" works, but it's very much over-simplified. Even so, it's a useful place to get people starting to think about things in the right way.

ColinWright 14y ago

It doesn't do it on the real numbers at all, it does it on a subset of rotations in R^3. You're right that these aren't "pieces" in the intuitive visual sense. That's why Feynman ridiculed the result, because in the initial description given to him he assumed physical pieces, not pairwise disjoint sets. http://news.ycombinator.com/item?id=411307 http://news.ycombinator.com/item?id=411307 There are similar paradoxical dissections of the plane: http://news.ycombinator.com/item?id=411637 http://news.ycombinator.com/item?id=411637

CJefferson 14y ago
You are right, I was oversimplifying perhaps too far!
- ColinWright 14y ago
  But it is a way in to understanding the way things work. It helps to have simpler situations that exhibit aspects of the whole thing. I remember being told about the library that has all finite books that can possibly be written. Every finite string of characters appears as a book. Obviously there are infinitely many, but we won't worry too much about that. Now we find that someone, overnight, stole all the books except for those that start with the letter "q". We're annoyed at first, but then someone suggests we just erase the initial "q" from all the remaining books. So we do that, and lo and behold! We have the full collection again! We even now have an extra book, one that's completely blank! It's not a paradox, but it is the way infinite things can work, and it shows how 1/26 of a collection can be the same as the whole collection. This is related to how the B-T "paradox" works, but it's very much over-simplified. Even so, it's a useful place to get people starting to think about things in the right way.