6 ms·
Known packings of equal circles in a circle
- raven105x 7y agoWould this not be an excellent design cue for friction-based water heaters?
- parsimo2010 7y agoIs 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?
- dfeojm-zlib 7y agoNewtonian packing of circles, e.g., Hungarians packing a Subaru for a summer road trip. :)
- HeWhoLurksLate 7y agoI don't even understand. What?
- lacker 7y agoNice. This will come in handy the next time I need to fit 1846 equal circles into a single larger circle.
- parsimo2010 7y agoMaybe you're being facetious, but for a CNC operator they might find it useful in an unusual machining job- they probably stick with a hexagonal lattice and rectangular stock most of the time. Or a chip foundry might find it useful if they are maximize yields in a wafer of silicon. It probably isn't immediately applicable to every single person's life but it might help some industries squeeze another 0.1% out of their production line. The free work presented here might cover a couple engineer's salaries once they find this website.
- mikepurvis 7y agoI feel like the more valuable thing would probably be the generalization of "fit as many of [irregular polygon] into the least number of length inches of material width X." OTOH, there are a lot of these math toys that start life in the abstract and then end up finding extremely practical applications down the line— pretty sure that was the case for a lot of the dusty corners of linear algebra until 3D graphics was suddenly a thing and it all became super relevant very quickly.
- terminalhealth 7y agoI personally just used this to drill a bunch of vent holes, though N = 37.
- terminalhealth 7y agoIt's interesting that beyond some number of circles the optimal solution likely always involves a regular equilateral grid in the center. Also interesting that there are seemingly no solutions without loose circles beyond 91.
- mikepurvis 7y agoAfter hitting a high enough size ratio, the spaces leftover when you inscribe a hexagon (built from small circles) inside the large circle are basically always going to require filling with loose circles. It's probably provable, but I'm not sure how you'd actually go about it.
- pavel_lishin 7y agoWhat do the colors indicate? http://hydra.nat.uni-magdeburg.de/packing/cci/d1.html http://hydra.nat.uni-magdeburg.de/packing/cci/d1.html I see orange, blue, purple and yellow on the page; what do they signify?
- Jabbles 7y agoThey correspond to the number of circles that this circle touches. The lines in the center of the circle show the directions.
- XaspR8d 7y agoAt first glance, blues seem to have 2 contact points and magentas have none, but the correspondence seems to break for yellow and orange, which each have several (overlapping) numbers of neighbors that they are used with.
- pavel_lishin 7y agoAh, I see it now. A magenta circle touches no other circles. A blue circle touches one other circle. An orange circle touches two other circles. A yellow circle touches three other circles.
- deleted 7y ago[deleted]
- dhritzkiv 7y agoThat's what I thought as well, but that theory doesn't appear to hold up. For example: http://hydra.nat.uni-magdeburg.de/packing/cci/cci14.html http://hydra.nat.uni-magdeburg.de/packing/cci/cci14.html
- deleted 7y ago[deleted]
- percentcer 7y ago
- 2bitencryption 7y agoI wonder, are there any patterns here for certain interesting mathematical sets of numbers, like primes, squares, etc? First glance doesn't show anything obvious, but I'm no mathematician.
- vesinisa 7y agoI find it most fascinating looking for the locally maximal densities. Starting from N=2, some arrangements always fall in a "satisfying" pattern and a locally optimal maximum density is achieved. "The sequence of N's that establish density records" link leads to an empty page, but this sequence is also known as OEIS A084644 "Best packings of m>1 equal circles into a larger circle setting a new density record", and starts with 2, 3, 4, 7, 19, 37, 55, 85, 121, 147, 148, 150, 151, 187. https://oeis.org/A084644 https://oeis.org/A084644 Look for example at N=1759: http://hydra.nat.uni-magdeburg.de/packing/cci/cci1759.html http://hydra.nat.uni-magdeburg.de/packing/cci/cci1759.html Compare it with N=1758, which has a slight "imperfection": http://hydra.nat.uni-magdeburg.de/packing/cci/cci1758.html http://hydra.nat.uni-magdeburg.de/packing/cci/cci1758.html Or with N=1760, which is too "tight" resulting in a worse density: http://hydra.nat.uni-magdeburg.de/packing/cci/cci1760.html http://hydra.nat.uni-magdeburg.de/packing/cci/cci1760.html
- madengr 7y agoThis is pretty important for modulation in digital communications. Interesting to see others than 2^N, and that 2^N are not square constellations, except for QPSK. Sometimes minimizing amplitude (envelope) variations is more important, or sometimes susceptibility to white noise or phase noise. I remember learning about N-dimensional sphere packing for coding.