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Known packings of equal circles in a circle (uni-magdeburg.de)
74 points by terminalhealth 76 days ago | hide | past | web | favorite | 35 comments

What do the colors indicate?


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

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).

Blue isn't only for N=2; it starts showing up again at N=104.

It appears that circles are colored blue if their number of contacts (with either other circles or the outside) is equal to 1 or 2. They're magenta if their number of contacts is zero.

I'm not sure what the distinction between orange and yellow is-- it doesn't seem to be strictly number of contacts, because I can see cases where both orange and yellow have four contacts.

Reread my answer for orange, I'm sure it is correct. To say it a different way: a circle X is coloured orange if X is touching a circle Y, and both X and Y also touch circle Z. Y and Z can either be small circles or the main outer circle.

The colours are about constraints. Orange is a locked constraint between 3 circles (inner and outer included).

If you think about it, N=2 is the only solution that the circles are constrained to touch in two places (and N=1 is weird because it touches in infinite places).

At N=104, the blue circles are surely a drawing fault, they are actually unconstrained and should be magenta. Shown by fact that those blue circles have a dot in the middle (not touching), but a neighbour circle has a line pointing to the blue circle (touching), which is contradictory.

For N=104, if you look at the PDF and zoom in, you can see the blue circles have a line rather than a dot. Admittedly, it is hard to see why the blue circles do not have enough freedom to move slightly away and become magenta (particularly the one on N=108).

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

At 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.

Ah, 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.

That'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

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.



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

That part of the FA you're quoting refers to the table, not the actual images of the circles; the part of the FA that I directly linked seems to imply something different.

Unless specific circles in the diagram are somehow linked to the specific authors?

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

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.

I 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.

I personally just used this to drill a bunch of vent holes, though N = 37.

It would make a cool series of posters.

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.

After 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.

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

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.

"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.



    = 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.

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.

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.

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.

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.

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

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

I don't even understand. What?

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