Orthorhombic sphere packings. II. Bivariant lattice complexes.

All homogeneous sphere packings and all interpenetrating layers of spheres were derived that refer to the 22 orthorhombic bivariant lattice complexes. In total, sphere packings of 90 different types have been found. Only for 47 of these types is the maximal inherent symmetry of their sphere packings orthorhombic.

Orthorhombic sphere packings. I. Invariant and univariant lattice complexes.

All homogeneous sphere packings and all interpenetrating sphere packings were derived that refer to the 6 invariant and the 11 univariant lattice complexes belonging to the orthorhombic crystal system. In total, sphere packings of 38 types have been found. Only for 17 types is the maximal inherent symmetry of their sphere packings orthorhombic.

Hexagonal and trigonal sphere packings. IV. Trivariant lattice complexes of trigonal space groups.

The 13 trivariant lattice complexes with trigonal symmetry are compatible with 218 types of homogeneous sphere packings, 7 types of interpenetrating sphere packings and one type of interpenetrating layers of spheres. Altogether, the lattice complexes with trigonal characteristic space group (with 0, 1, 2 or 3 degrees of freedom) give rise to 225 types of sphere packing. 110 of these types are compatible exclusively with one of the 13 trivariant lattice complexes, 31 in addition with some of the invariant, univariant or bivariant lattice complexes, whereas 6 types occur solely in such a lattice complex.

Tetragonal sphere packings: minimal densities and subunits.

For all 382 types of homogeneous sphere packings with tetragonal symmetry, the minimal sphere-packing densities have been calculated. The tabulated coordinates allow the graphic representation of a sample packing for each type. 1- and 2-periodic subunits of these sphere packings are listed in addition.

On the density of homogeneous sphere packings.

For some types of sphere packing with typical one- and two-dimensional parameter regions, the sphere-packing density as a function of the free parameters is discussed. In addition, some sphere-packing types with extraordinary density properties are presented. Until now, it was generally assumed that sphere packings with minimal density are also those of highest inherent symmetry.

Hexagonal and trigonal sphere packings. III. Trivariant lattice complexes of hexagonal space groups.

All types of homogeneous sphere packing and interpenetrating sphere packings and layers were derived that correspond to point configurations of the 15 trivariant hexagonal lattice complexes. The respective sphere packings are assigned to 147 types. In total, sphere packings of 170 types can be realized with hexagonal symmetry.

Minimal densities of cubic sphere-packing types.

The minimal sphere-packing densities have been calculated for all 199 types of homogeneous sphere packings with cubic symmetry. The tabulated coordinates allow the graphic representation of each type.

Exceptional properties of some sphere packings in the general position of P6(2)22.

Three types of sphere packing are described which show properties that have never been observed before: a certain set of generating symmetry operations corresponds to a parameter range in P6(2)22 that is not simply connected, but disintegrates into two disjoint non-congruent regions; the minimal sphere-packing density is different for these two regions; two sphere packings from different regions cannot be deformed into each other without opening sphere contacts although their sphere-packing graphs are isomorphic in the graph-theoretical sense. Two heterogeneous crystal nets with different symmetry described by Delgado-Friedrichs & O'Keeffe [Acta Cryst. (2003), A59, 351-360] show a similar behaviour.

Hexagonal and trigonal sphere packings. II. Bivariant lattice complexes.

All homogeneous sphere packings were derived which correspond to point configurations of the 26 bivariant lattice complexes belonging to the hexagonal crystal family. They may be assigned to 109 sphere-packing types. Among these, there is a type of sphere packing with contact number 10 that was not described before.

Hexagonal and trigonal sphere packings. I. Invariant and univariant lattice complexes.

All homogeneous sphere packings and all interpenetrating sphere packings have been derived that refer to the seven invariant and the 23 univariant lattice complexes belonging to the hexagonal crystal family. The respective sphere packings may be assigned to 66 types. In addition, one case of interpenetrating sphere packings was found.

Homogeneous sphere packings with triclinic symmetry.

All homogeneous sphere packings with triclinic symmetry have been derived by studying the characteristic Wyckoff positions P -1 1a and P -1 2i of the two triclinic lattice complexes. These sphere packings belong to 30 different types. Only one type exists that has exclusively triclinic sphere packings and no higher-symmetry ones.

Minimal surfaces with self-intersections along straight lines. I. Derivation and properties.

A special kind of three-periodic minimal surface has been studied, namely surfaces that are generated from disc-like-spanned skew polygons and that intersect themselves exclusively along straight lines. A new procedure for their derivation is introduced in this paper. Several properties of each such surface may be deduced from its generating polygon: the full symmetry group of the surface, its orientability, the symmetry group of the oriented surface, the pattern of self-intersections, the branch points of the surface, the symmetry and periodicity of the spatial subunits demarcated by the surface, and the Euler characteristics both of the surface and of the spatial subunits.