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Comment by pavel_lishin | Hacker News Reader

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?

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.

FTFA:

Legend:

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?

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