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).
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.
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.
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?
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.
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
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.
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.
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.
http://hydra.nat.uni-magdeburg.de/packing/cci/d1.html
I see orange, blue, purple and yellow on the page; what do they signify?