Abstract

Assuming that the densest packings of ellipses or ellipsoids are only those which result from the closest packings of circles or spheres respectively by an affine transformation, 2 different densest packings of ellipses and 13 different densest packings of ellipsoids are derived: Two packings of ellipses (C_2v^IV and C₂^I), seven packings of ellipsoids (D_4h¹⁷, D_3d⁵, D_2h²³, D_2h²⁵, two × C_2h³, C_i¹) and six packings of ellipsoids (D_6h⁴, D_2h¹⁷, C_2h², C_2h³, C_2h⁶, C_i¹). They are related to the closest packing of circles (C_6v^I), the c.c.p. of spheres (O_h⁵) and the h.c.p. of spheres (D_6h⁴) respectively. These densest packings of ellipses and ellipsoids (which are parallel arrangements) have the following characteristics:

1. they have the same density as the corresponding circle and sphere packing (s) (ϱ = [unk] = 0.9069… in the plane and ϱ = [unk] = 0.7404… in space);

2. they have the same number of contacts as the corresponding circle and sphere packing(s): 6 resp. 12;

3. their space groups are subgroups of tho corresponding circle and sphere packings (C_6v^I resp. O_h⁵ and D_6h⁴), with the same centers of symmetry.