Known packings of equal circles in a circle

What do the colors indicate?<p><a href="http:&#x2F;&#x2F;hydra.nat.uni-magdeburg.de&#x2F;packing&#x2F;cci&#x2F;d1.html" rel="nofollow">http:&#x2F;&#x2F;hydra.nat.uni-magdeburg.de&#x2F;packing&#x2F;cci&#x2F;d1.html</a><p>I see orange, blue, purple and yellow on the page; what do they signify?

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

I find it most fascinating looking for the locally maximal densities. Starting from N=2, some arrangements always fall in a &quot;satisfying&quot; pattern and a locally optimal maximum density is achieved.<p>&quot;The sequence of N&#x27;s that establish density records&quot; link leads to an empty page, but this sequence is also known as OEIS A084644 &quot;Best packings of m&gt;1 equal circles into a larger circle setting a new density record&quot;, and starts with 2, 3, 4, 7, 19, 37, 55, 85, 121, 147, 148, 150, 151, 187. <a href="https:&#x2F;&#x2F;oeis.org&#x2F;A084644" rel="nofollow">https:&#x2F;&#x2F;oeis.org&#x2F;A084644</a><p>Look for example at N=1759: <a href="http:&#x2F;&#x2F;hydra.nat.uni-magdeburg.de&#x2F;packing&#x2F;cci&#x2F;cci1759.html" rel="nofollow">http:&#x2F;&#x2F;hydra.nat.uni-magdeburg.de&#x2F;packing&#x2F;cci&#x2F;cci1759.html</a><p>Compare it with N=1758, which has a slight &quot;imperfection&quot;: <a href="http:&#x2F;&#x2F;hydra.nat.uni-magdeburg.de&#x2F;packing&#x2F;cci&#x2F;cci1758.html" rel="nofollow">http:&#x2F;&#x2F;hydra.nat.uni-magdeburg.de&#x2F;packing&#x2F;cci&#x2F;cci1758.html</a><p>Or with N=1760, which is too &quot;tight&quot; resulting in a worse density: <a href="http:&#x2F;&#x2F;hydra.nat.uni-magdeburg.de&#x2F;packing&#x2F;cci&#x2F;cci1760.html" rel="nofollow">http:&#x2F;&#x2F;hydra.nat.uni-magdeburg.de&#x2F;packing&#x2F;cci&#x2F;cci1760.html</a>

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&#x27;ve probably got the best possible packing for small N, but the larger numbers look like there may be room for improvement.

I wonder, are there any patterns here for certain interesting mathematical sets of numbers, like primes, squares, etc?<p>First glance doesn&#x27;t show anything obvious, but I&#x27;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.<p>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. :)