On packing axisymmetric bodies in 3D 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
 A: 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.
A: As a supplementary answer, here are some notes on the 2D analog mentioned in the original post.
The conjecture is false for nonconvex bodies:

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.)
A: 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.
