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Consider any 3D body with an axis of rotational symmetry (e.g. cone, cylinder...) and packing the 3d space efficiently with infinitely many copies of this body. Is the following claim valid?

Claim: The densest packing with any such body is necessarily such that all units are aligned along or opposite to the same direction

Proving such a claim will greatly limit the possibilities that need to be considered to find the densest packing. The 2D analog of the claim above would involve bodies with a reflection symmetry.

Note: This question was recorded at https://nandacumar.blogspot.com/2019/02/on-packing-with-axi-symmetric-bodies.html

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  • $\begingroup$ I'm pretty sure in this general form the claim is false. You can imagine lots of bodies with rotational symmetries that are non-convex, maybe, and for which a packing with unaligned copies is more efficient. This is not an answer, but more like a hunch. $\endgroup$
    – Beni Bogosel
    Commented Jul 15, 2021 at 19:43
  • $\begingroup$ Thanks. Hopefully, the claim may be more interesting when restricted to convex bodies - and finding counters harder. $\endgroup$
    – Nandakumar R
    Commented Jul 15, 2021 at 19:52
  • $\begingroup$ Are you allowing only rotations and translations of the original body, or also dilations? Are you allowing infinite bodies or not? And will you give an example of the packings you have in mind? $\endgroup$
    – user44143
    Commented Jul 17, 2021 at 4:04
  • $\begingroup$ I was considering packing non-dilated copies of a given finite axisymmetric body in 3D. As an example of a question on densest packings, one could mention the Kepler conjecture. $\endgroup$
    – Nandakumar R
    Commented Jul 17, 2021 at 17:38
  • $\begingroup$ Do you require that the shape is invariant under rotation at any angle, or just at some nontrivial angle? (For instance, does a cube count?) $\endgroup$
    – RavenclawPrefect
    Commented Jul 23, 2021 at 21:10

3 Answers 3

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Your claim is false for axially-symmetric ellipsoids: when restricted to all have their axis of symmetry in the same direction, they cannot pack more densely than spheres (in terms of volume fraction). However, they can in fact pack more densely than spheres, as discussed in Donev et al and references therein.

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  • $\begingroup$ Thanks! The reference says: "By inserting very elongated ellipsoids into cylindrical void channels passing thru the ellipsoidal analogs of the densest ordered sphere packings (an affinely deformed face-centered cubic or hexagonal close packed lattice), congruent ellipsoid packings can be made with density > 0.7405 (density of affinely deformed fcc or hcp) (for) thin ellipsoids of revolution" - and these inserted ellipsoids will have a different orientation from the ones in the 'main lattice'. Guess if the orientations of all units should be same, one can't do better than fcc or hcp. $\endgroup$
    – Nandakumar R
    Commented Jul 24, 2021 at 19:57
  • $\begingroup$ @NandakumarR That's right. If orientation of all units is the same then an affine transformation makes the packing into a sphere packing so can't have larger density than fcc. The paper I referenced shows that even ellipsoids that are nearly spherical can pack more densely than spheres. $\endgroup$
    – Yoav Kallus
    Commented Jul 25, 2021 at 20:23
  • $\begingroup$ Guess this shows how special Kepler conjecture is! $\endgroup$
    – Nandakumar R
    Commented Jul 27, 2021 at 17:51
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As a supplementary answer, here are some notes on the 2D analog mentioned in the original post.

The conjecture is false for nonconvex bodies:

enter image description here

The above shape can tile the plane without gaps when rotated, but allowing only $180^\circ$ rotations does not permit this.

However, if we restrict to centrally symmetric convex shapes (a slightly different condition than reflectional symmetry, but another reasonable analog of rotational symmetry for 2D bodies) then it is true: a theorem of L. Fejes Tóth from 1949 states that any centrally-symmetric 2D body has a packing density equal to its translational packing denity, as shown in paper Some packing and covering theorems. (He mentions in footnote 8 that this result may have been independently given in an October 1949 lecture of K. Mahler.)

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There is another, relevant and older publication on this very topic, with a construction much similar to the one in the article cited by Yoav Callus in his answer above:

Bezdek, A., Kuperberg, W., Packing Euclidean space with congruent cylinders and with congruent ellipsoids. Applied geometry and discrete mathematics, 71–80, DIMACS Ser. Discrete Math. Theoret. Comput. Sci., 4, Amer. Math. Soc., Providence, RI, 1991, MR1116339.

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  • $\begingroup$ Thank you! Since the 'basic' claim that all units have one of two mutually opposite orientations is clearly invalid, a natural 'relaxation' would be to ask if there is an upper bound on the number of orientations in the best pack of any 3d body with rotational symmetry. $\endgroup$
    – Nandakumar R
    Commented Sep 17, 2021 at 5:58

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