Take a (solid) sphere ... there are uncountably infinitely many points.
Now for each point colour it read, green, blue, yellow, or purple. Do this completely randomly.
Then every point will have nearby, arbitrarily close, other points of every colour. The colours will be deeply intertwungle, with the points of each colour just being a "cloud", spread throughout the entire volume.
This is a better mental image of what the "pieces" are like.
Now, if you do this colouring in a very, very special way, the red points can be rotated around to match exactly the green points. Well, OK, so the green points are a rotation of the red points. Not a problem.
But you can also arrange it so that if you rotate the red points a different way then they will exactly match the blue points. Well, OK, so the blue points are also a rotation of the red points.
The utterly, utterly bizarre result is that the above can be true, and we can also have a third rotation of the red points that will match all the green points and the blue points AT THE SAME TIME!
At its heart, that how the B-T theorem works. There are details, and arranging that this happens in also non-trivial, but at its heart, this is what's happening.
The purpose of the theorem is to show that the concept of "volume" cannot be applied to arbitrary sets of points in 3D. If you want more details about that, I wrote a blog post about this many years ago. It was my project for my BSc (Hons) way back in 1982.
Comments
The "pieces" aren't pieces such as you picture.
Take a (solid) sphere ... there are uncountably infinitely many points.
Now for each point colour it read, green, blue, yellow, or purple. Do this completely randomly.
Then every point will have nearby, arbitrarily close, other points of every colour. The colours will be deeply intertwungle, with the points of each colour just being a "cloud", spread throughout the entire volume.
This is a better mental image of what the "pieces" are like.
Now, if you do this colouring in a very, very special way, the red points can be rotated around to match exactly the green points. Well, OK, so the green points are a rotation of the red points. Not a problem.
But you can also arrange it so that if you rotate the red points a different way then they will exactly match the blue points. Well, OK, so the blue points are also a rotation of the red points.
The utterly, utterly bizarre result is that the above can be true, and we can also have a third rotation of the red points that will match all the green points and the blue points AT THE SAME TIME!
At its heart, that how the B-T theorem works. There are details, and arranging that this happens in also non-trivial, but at its heart, this is what's happening.
The purpose of the theorem is to show that the concept of "volume" cannot be applied to arbitrary sets of points in 3D. If you want more details about that, I wrote a blog post about this many years ago. It was my project for my BSc (Hons) way back in 1982.