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>



On packing axisymmetric bodies in 3D
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](https://dx.doi.org/10.1103/PhysRevLett.92.255506) and references therein.


