An interpenetrating sphere packing is defined as a set of crystallographically equivalent spheres, which form two or more sphere packings without mutual contact. All types of interpenetrating sphere packings with cubic symmetry have been deduced, using the known symmetry regions for cubic space groups. These 39 types show 12 different kinds of interpenetration. Different types are called related if an interpenetrating sphere packing of one type can be generated from another one by omitting contacts or spheres. All such relations are tabulated.

The Euclidean normalizers of space groups form the appropriate mathematical tool for several problems treated independently by crystallographers in the past, e.g. the comparison, the classification and the standardized description of crystal structures. Explicit tables are presented that enable the user to handle Euclidean normalizers in an easy way and, especially, to calculate all descriptions of a crystal structure compatible with a chosen space-group setting. The use of the tables is illustrated by different examples, and the role of Euclidean normalizers for crystal-structure determination is discussed.

Three new families of minimal balance surfaces have been derived. For this a new kind of surface patch, i.e. branched catenoid, has been used. A concave polygon with one point of self-contact and a convex polygon are the two generating circuits of such a minimal balance surface.

The genus of a three-periodic intersection-free surface in R³ refers to a primitive unit cell of its symmetry group. Two procedures for the calculation of the genus are described: (1) by means of labyrinth graphs; (2) via the Euler characteristic derived from a tiling on the surface. In both cases new formulae based on crystallographic concepts are given. For all known minimal balance surfaces the genera and the labyrinth graphs are tabulated.

Two new families of minimal balance surfaces are described. Their surface patches are not finite but have the shape of infinite strips. Such a strip is bounded by two congruent zigzag lines in one case or by a zigzag line and a meander line in the other case. In addition, certain minimal balance surfaces derived before with the aid of finite surface patches can also be generated from infinite strip-like surface patches.

Most lattice complexes with less than three degrees of freedom are contained as limiting complexes within several comprehensive complexes corresponding to general Wyckoff positions. There exist, however, 21 exceptional cases: 3 lattice complexes without any and 18 lattice complexes with exactly one such comprehensive complex each. They are listed together with their comprehensive complexes. This information may be used to restrict geometrical investigations of all point configurations to general positions of space groups.

Eight intermetallic compounds described so far by clusters are compared. It is shown that a common framework exists for all these crystal structures. The voids within this common part are filled in different ways. Moreover, for each of these crystal structures all atoms together form a three-dimensional connected framework. Therefore, the framework description seems to be more adequate than the cluster description.

For all space groups the groups of inner automorphisms are given. They are isomorphic with groups of motions, but fall into four sets according to their dimension. In the triclinic and monoclinic systems, however, the corresponding groups of all automorphisms cannot be represented by groups of motions. These groups of automorphisms therefore are only tabulated for the other cases. The relation between groups of automorphisms and Cheshire groups is discussed. By means of automorphisms sets of equivalent points are combined to "Konfigurationslagen'. For these complete tables are included.

Eight new families of minimal balance surfaces are described. Their surface patches belong to a new kind, called multiple catenoids. The generating circuits of such a minimal surface are two congruent concave polygons with one point of self-contact each. The new minimal balance surfaces are complementary to other minimal balance surfaces which are built up from catenoid-like surface patches and have been known before.

The way in which Schoen [Infinite Periodic Minimal Surfaces Without Self-intersections (1970), NASA Tech. Note No. D-5541] derived a simply connected surface patch for a C(H) surface cannot be generalized. One may, however, subdivide a C(H) surface into larger patches that are not simply connected. Surface patches of analogous shape give rise to five families of minimal balance surfaces unknown so far: tetragonally and orthorhombically distorted C(P) surfaces, surfaces complementary to Schoen's R2 and R3 surfaces with genus 25 and 37, respectively, and orthorhombic surfaces of a fifth family with genus 5 that are also complementary to oP surfaces.

The symmetry conditions for flat points of minimal surfaces have been studied in relation to the order β of points on such surfaces. Using symmetry aspects, a set of rules for the derivation of fiat points have been developed. By means of these rules the flat points for the 45 families of minimal balance surfaces known so far have been determined. As a check for completeness the relation between the genus of a minimal surface and the orders of its flat points has been used.

A listing of the Euclidean normalizers for triclinic and monoclinic space groups having translation lattices with specialized metric is given. These normalizers have not been included in previous tabulations. For convenience, in the case of monoclinic space groups, only the second setting (c-axis unique) is considered and the metric of the cells is restricted within certain limits which warrant that all specialized cases and all cell choices according to International Tables for Crystallography [(1987) Dordrecht: Kluwer] are included.