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Does there exist a finite set of radii such that some aperiodic packing of the plane by disks of those radii is believed to achieve the maximal packing density (not achieved by any periodic packing)?

I would also be interested in a nonconstructive proof that such a set of radii must exist.

Added on 6/4/20: I just discovered https://arxiv.org/pdf/1111.4917.pdf ("Densest binary sphere packings" by Hopkins, Stillinger, and Torquato) which asserts "In $R^2$, periodic, quasicrystalline, and directionally periodic structures can all be found among the putative densest binary disk packings [38, 50–52]." However I have not yet looked at the references for details. I'll update this post again if I learn more.

[Note to administrators: I wanted to add “aperiodic” or “quasicrystal” or something like that as a tag; I settled for “almost-periodic-function”, but please re-tag as appropriate.]

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    $\begingroup$ +1. It's interesting whether this can happen even for only two different radii. $\endgroup$
    – Josiah Park
    Commented Jun 2, 2020 at 18:16
  • $\begingroup$ Just checking...do you really mean aperiodic or do you perhaps mean non-periodic? $\endgroup$
    – Timothy Chow
    Commented Jun 4, 2020 at 16:21
  • $\begingroup$ I mean aperiodic (like a Penrose tiling), as opposed to nonperiodic (like a hexagonal close-packing of unit disks with tiny disks of radius 1/100 rattling around in the interstices in a nonperiodic fashion). $\endgroup$
    – James Propp
    Commented Jun 4, 2020 at 21:42
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    $\begingroup$ One thing to note as you look at the references you cited, is that usually the proportion of discs of different sizes is also perscribed, and I believe the relevant question then is whether the optimal density can always be achieved by a mixture of one or more periodic structures. $\endgroup$
    – Yoav Kallus
    Commented Jun 4, 2020 at 22:27

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