4 ms·
Is there a proven bound so we know when the best known packing is the best possible packing? The lower numbers look very tidy and we've probably got the best p
by parsimo2010 7y ago
Is there a proven bound so we know when the best known packing is the best possible packing? The lower numbers look very tidy and we've probably got the best possible packing for small N, but the larger numbers look like there may be room for improvement.
- sp332 7y agoProven optimal packings are indicated by a radius in bold face type.
- thefifthsetpin 7y agoWhich they then served as a blurry (in my browser, at least) gif. I have no idea which #'s are bold.
- parsimo2010 7y agoSo is it that we don't know have a bound for generic N, or that we know a bound but haven't found it for N=14-18, >20? I'm sure we know the bound exists, but I'm more interested in whether we've found a closed form (or even just analytic) way to express the bound.
- alexis_fr 7y agoThere is at least one bound: When the unused surface is smaller than 1 disc is an absolute bound. Also, when all discs are touching 3-by-3, we can’t extract more space.
- 1wd 7y ago"David W. Cantrell gives in [24] an astonishing new conjectured upper bound for R/r, namely R/r <= 1 + (sqrt((4ρ-1)^2 + 16ρ(N-1)) - 1) / (4ρ) with ρ = Pi/(2*sqrt(3)). ..." But the references section seems to be missing entry [24] for some reason.
- jandrese 7y agoHowever: ratio = 1/radius; an orange field means that David W. Cantrell's conjectured upper bound is violated Seems like the conjecture didn't hold up in practice.
- 1wd 7y agoIs it proven to be violated? Or just currently "violated" by the best known (but not necessarily optimal) approach?