Covering and packing
Aug. 23rd, 2021 12:54 amSo I was talking to Drake and he was showing me some questions he had posted on MathOverflow and math.stackexchange. And one of them was this one.
And it was like -- huh, for some reason you don't see this sort of problem stated very often?
Like, packing figures in the plane (or higher-dimensional spaces), sure, you see packing problems all the time. But covering problems? Where the copies can overlap, and you have to cover the whole plane (or space), and the goal is essentially to minimize the overlap? (Really, to minimize the area used, counted with multiplicity.) For whatever reason you don't see that stated!
This is surprising because like... these are dual problems, right?. Packing and covering, these are dual problems. And yet somehow I never before it never occurred to me as strange that when things get geometric like this, you see people talking about packing problems, but not covering problems.
Drake said you do sometimes see covering problems, but rarely enough that they're basically the just the simplest cases. Not the wide variety you see with packing problems.
This seems like a distinct omission from the body of known mathematics. Geometers and combinatorialists (or whoever studies this sort of thing)... you need to get on this!
And it was like -- huh, for some reason you don't see this sort of problem stated very often?
Like, packing figures in the plane (or higher-dimensional spaces), sure, you see packing problems all the time. But covering problems? Where the copies can overlap, and you have to cover the whole plane (or space), and the goal is essentially to minimize the overlap? (Really, to minimize the area used, counted with multiplicity.) For whatever reason you don't see that stated!
This is surprising because like... these are dual problems, right?. Packing and covering, these are dual problems. And yet somehow I never before it never occurred to me as strange that when things get geometric like this, you see people talking about packing problems, but not covering problems.
Drake said you do sometimes see covering problems, but rarely enough that they're basically the just the simplest cases. Not the wide variety you see with packing problems.
This seems like a distinct omission from the body of known mathematics. Geometers and combinatorialists (or whoever studies this sort of thing)... you need to get on this!
