Known packings of equal circles in a circle

[pavel_lishin]

What do the colors indicate?

http://hydra.nat.uni-magdeburg.de/packing/cci/d1.html

I see orange, blue, purple and yellow on the page; what do they signify?

[robocat]

Orange is three circles all touching each other (hexagonal close pack), or two circles touching each other and the outside (I have reedited this sentence for clarity).

Magenta is unconstrained.

Blue is special case that can only exist for N=2.

The rest are yellow.

The number of lines in the middle show the number of touchs (constraints, 0 to 6).

Not sure how multiple solutions are shown (e.g. N=6 has two solutions).

[Jabbles]

They correspond to the number of circles that this circle touches. The lines in the center of the circle show the directions.

[percentcer]

The pink ones are rattlers, that's all I could figure out. I thought orange vs yellow implied something about how tightly packed they were but that doesn't seem to hold up.

[sizzzzlerz]

FTFA:

Legend:

N the number of circles; colors correspond to active researchers in the past, see "References" at the bottom of the page

[lacker]

Nice. This will come in handy the next time I need to fit 1846 equal circles into a single larger circle.

[parsimo2010]

Maybe 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.

[H8crilA]

It would make a cool series of posters.

[terminalhealth]

It'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.

[vesinisa]

I 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

Look for example at N=1759: 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

Or with N=1760, which is too "tight" resulting in a worse density: http://hydra.nat.uni-magdeburg.de/packing/cci/cci1760.html

[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.

[1wd]

"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.

[sp332]

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

[2bitencryption]

I 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.

[madengr]

This 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.

[raven105x]

Would this not be an excellent design cue for friction-based water heaters?

[dfeojm-zlib]

Newtonian packing of circles, e.g., Hungarians packing a Subaru for a summer road trip. :)

[HeWhoLurksLate]

I don't even understand. What?