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Comment by parsimo2010 | Hacker News Reader

parsimo2010

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.

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"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.

However:

  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.

Is it proven to be violated? Or just currently "violated" by the best known (but not necessarily optimal) approach?

Proven optimal packings are indicated by a radius in bold face type.

Which they then served as a blurry (in my browser, at least) gif. I have no idea which #'s are bold.

It's not referring to the images linked at the top, but the table at the bottom of the page. It's not an image.

The PDFs are vector image based.

So 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.

There 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.

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