A grasshopper lands at a random point on a planar lawn of area one. It then jumps once, a fixed distance 𝑑, in a random direction. What shape should the lawn be to maximize the chance that the grasshopper lands on the lawn again?
Surprisingly, the lawn should never be shaped like a disk! Here's what it should look like for various choices of 𝑑. For larger values of 𝑑 it gets even weirder.
@drmorr - I don't know if anyone has that intuition for small 𝑑, which is the case my the pictures in my post showed. But for large 𝑑 it's obvious that a jump of length 𝑑 will completely throw you out of a disk of area 1, so the best shape can't be a disk.
@johncarlosbaez@drmorr As for intuition I can be kinda handwavy for small 𝑑 < 1/√π ~ 0.56.
Say the grasshopper's first leap lands exactly in the middle of the disk. As @johncarlosbaez points out when 𝑑 is minutely above the radius of a disk of area 1 (say, .57), you can see how you might want to modulate the boundary in a periodic way to cover greater distance jumps, sometimes. Hence the cogs.
As the grasshopper lands randomly rather than exactly in the center, that handwavy intuition extends to distances less than 0.56.
I have no feel for the bladed or striped regimes, though. Neat paper.
@johncarlosbaez@esoterica The original post was about Bell's inequalities. Someone with more math genes than me should comment on my comment once, or just steal the idea outright. Do you care to comment Mr. Baez?
"I suspect that the only thing Bell's inequalities say is that you cannot fix an indeterminate value. Like when a coin is spinning during a toss, you cannot fix the outcome to either heads or tails, only measure after it lands."
I suggest that a correlation between indefinite states, measurements described by sampling probability distributions, is different from a correlation between definite states, or plain measurements on a shared state described by a hidden variable.
@johncarlosbaez@esoterica One last remark since this didn't let me go the last days and I think my last comment made it worse.
I think what Bell did is that he tried to give a QM system (of particles) a definite value by chosing a hidden variable. And it shouldn't surprise anyone, especially those working with QM, that that doesn't work out.
That's just a hunch but it leaves a nagging suspicion that QM is local.
@johncarlosbaez@esoterica
It seems like the optimal shape has really small deformations until a jump in the number of cogs. What happens at the transition? Is there a continuum of optimal shapes deforming from (n+1)-cogs to n-cogs or does it jump discontinuously? From glancing at the paper, it seems discontinuous bet I am also unsure if they really tried to pin down the (d_\text{transition}) precisely
@MaloTarpin - I believe there are discontinuous jumps. When 𝑑 increases further there is an even more dramatic discontinuous transition which they study in detail.
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