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It is not so surprising that the problem of Circle packing in a square is a chaotic and often-unpredictable problem. However, after looking over the data on hydra.nat.uni-magdeburg.de, we find quite the anomaly.

Here is a graph of the first ~1000 densities of the best known solutions.

[Circle packing in a square graph1

The graph is perhaps unsurprisingly chaotic, beit with some patterns, of course. However, Something incredibly unusual / significant seems to happen at number 992.

Below I've added its solution, compared to known neighboring solutions.

Circle packing in a square graph, 991, 992, 998

Noticeably, it's rather regular, especially compared to it's neighboring solutions. This is likely the cause for the sudden change in density. Of course, the exact densities are subject to change, as better solutions are found. But even if new solutions are found at and near 992, I seriously doubt this anomaly will go away, given its relatively huge gap in the sequence, and the already enormous computation done over the years that went into this data. So my wondering is,

What is so special about 992 and its local neighborhood of solutions, that allows for it to have such a regular (yet inefficient) solution, and drop in density? Is it as simple as the fact that we haven't found a better solution, up to par with the others? Or is it just special, given that it's of the form $n(n+1)$, or of some other "nice" property?

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  • $\begingroup$ I've added the soft-question tag, given that my questions are rather qualitative, since it's not so trivial to quantify the nature of this anomaly. $\endgroup$
    – Graviton
    Commented Nov 27, 2022 at 3:21
  • $\begingroup$ That is simply the best KNOWN solution for 992 circles packed in a square. That does not discount the reality that there exists a way to pack 992 circles in a more intricate way with a higher density than is shown. $\endgroup$
    – Will Sherwood
    Commented Nov 27, 2022 at 3:21
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    $\begingroup$ I have no idea, I just want to comment because this is very beautiful $\endgroup$
    – Alborz
    Commented Nov 27, 2022 at 3:23
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    $\begingroup$ @Graviton That's true for the table as a whole, but I think that 992 has not been updated since its introduction in the May 2010 entry: "Totally 584 new packings in the range N = 507 ... 5000 by Eckard Specht [31]. These packings are strict regular lattice packings, many of them can be improved" $\endgroup$
    – Misha Lavrov
    Commented Nov 27, 2022 at 3:33
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    $\begingroup$ I would take the results at packomania with a grain of salt. They have wrong results for even simple things, for example the densest arrangement of four circles in a regular hexagon (there should be two opposite circles each tangent to two sides). $\endgroup$
    – Dan
    Commented Jan 11, 2023 at 4:17

1 Answer 1

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It appears that the table contains errors (or at least obviously inferior solutions). The maximum radius should never increase, as it does from $n=990$ to $n=991$:

990 0.016463423221309863655952299335
991 0.016714637930021076293666393417

You can trivially improve the given $n=990$ solution by arbitrarily removing any circle from the $n=991$ solution, preserving the larger radius.

And again from $n=992$ to $n=998$:

992 0.015873015873015873015873015873
998 0.016670442622470923459332814794

You can trivially improve the given $n=992$ solution by arbitrarily removing $6$ circles from the $n=998$ solution, preserving the larger radius.

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    $\begingroup$ Ah, I was hoping for a deep mystery! Alas this does explain the situation. Thanks for looking into it. A shame that errors like these creep in, but not unexpected. $\endgroup$
    – Graviton
    Commented Nov 27, 2022 at 3:52

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