
Known packings of equal circles in a circle - terminalhealth
http://hydra.nat.uni-magdeburg.de/packing/cci/cci.html
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pavel_lishin
What 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?

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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).

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JonathonW
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.

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robocat
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.

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showdead
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).

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

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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.

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mikepurvis
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.

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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](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)

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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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sp332
_Proven optimal packings are indicated by a radius in bold face type._

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

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sp332
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.

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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.

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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.

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raven105x
Would this not be an excellent design cue for friction-based water heaters?

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

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HeWhoLurksLate
I don't even understand. What?

