A needle (a segment) lies on a plane. One can rotate it 45 degree round any of its endpoints. Is it possible that after several rotations the needle returns to initial position with the endpoints interchanged?
Consider one particular series of moves.
wlog, the length is 1 unit with one end marked "A" at (0,0) and the other marked "B" at (1,0)
Take it through 4 counterclockwise moves (letter indicates pivot point): A,B,A,B.
We started at A(0,0)---B(1,0)
and ended at B(0,√2 - 1)---A(1,√2 - 1).
The positions are reversed as requested, but the needle has shifted 'North' by √2 - 1 units, which is an irrational amount.
I therefore speculate without formal proof that for the parity of the needle to be reversed and still be horizontal, the needle must be shifted in the xy plane by some irrational amount in some direction. Undoing that same irrational amount in the reverse direction to return to the original position will necessarily flip the parity again back to the original parity. So, no, the endpoints cannot be interchanged.
https://www.desmos.com/calculator/ywo9ior7zt
This problem reminded me of "Moving Furniture", posted 20 years ago.
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Posted by Larry
on 2024-10-03 12:30:42 |
Speculation without proof
