Tables for
Volume A
Space-group symmetry
Edited by Th. Hahn

International Tables for Crystallography (2006). Vol. A, ch. 14.3, p. 873

Section 14.3.1. Geometrical properties of point configurations

W. Fischera and E. Kocha*

aInstitut für Mineralogie, Petrologie und Kristallographie, Philipps-Universität, D-35032 Marburg, Germany
Correspondence e-mail:

14.3.1. Geometrical properties of point configurations

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To study the geometrical properties of all point configurations in three-dimensional space, it is not necessary to consider all Wyckoff positions of the space groups or all 1128 types of Wyckoff set. Instead, one may restrict the investigations to the characteristic Wyckoff positions of the 402 lattice complexes. The results can then be transferred to all noncharacteristic Wyckoff positions of the lattice complexes, as listed in Tables[link] and[link] .

The determination of all types of sphere packings with cubic or tetragonal symmetry forms an example for this kind of procedure (Fischer, 1973[link], 1974[link], 1991a[link],b[link], 1993)[link]. The cubic lattice complex I4xxx, for example, allows two types of sphere packings within its characteristic Wyckoff position [I\bar{4}3m] 8c xxx .3m, namely [9/3/c2] for [x = 3/16] and [6/4/c1] for [{3/16} \lt x \lt {1 \over 4}] (cf. Fischer, 1973)[link]. Ag3PO4 crystallizes with symmetry [P\bar{4}3n] (Deschizeaux-Cheruy et al., 1982[link]) and the oxygen atoms occupy Wyckoff position 8e xxx .3., which also belongs to I4xxx. Comparison of the coordinate parameter [x = 0.1491] for the oxygen atoms with the sphere-packing parameters listed for [I\bar{4}3]m c shows directly that the oxygen arrangement in this crystal structure does not form a sphere packing.

Other examples for this approach are the derivation of crystal potentials (Naor, 1958[link]), of coordinate restrictions in crystal structures (Smirnova, 1962[link]), of Patterson diagrams (Koch & Hellner, 1971[link]), of Dirichlet domains (Koch, 1973[link], 1984[link]) and of sphere packings for subperiodic groups (Koch & Fischer, 1978[link]).

The 30 lattice complexes in two-dimensional space correspond uniquely to the `henomeric types of dot pattern' introduced by Grünbaum and Shephard (cf. e.g. Grünbaum & Shephard, 1981[link]; Grünbaum, 1983[link]).


Deschizeaux-Cheruy, M. N., Aubert, J. J., Joubert, J. C., Capponi, J. J. & Vincent, H. (1982). Relation entre structure et conductivité ionique basse temperature de Ag3PO4. Solid State Ionics, 7, 171–176.
Fischer, W. (1973). Existenzbedingungen homogener Kugelpackungen zu kubischen Gitterkomplexen mit weniger als drei Freiheitsgraden. Z. Kristallogr. 138, 129–146.
Fischer, W. (1974). Existenzbedingungen homogener Kugelpackungen zu kubischen Gitterkomplexen mit drei Freiheitsgraden. Z. Kristallogr. 140, 50–74.
Fischer, W. (1991a). Tetragonal sphere packings. I. Lattice complexes with zero or one degree of freedom. Z. Kristallogr. 194, 67–85.
Fischer, W. (1991b). Tetragonal sphere packings. II. Lattice complexes with two degrees of freedom. Z. Kristallogr. 194, 87–110.
Fischer, W. (1993). Tetragonal sphere packings. III. Lattice complexes with three degrees of freedom. Z. Kristallogr. 205, 9–26.
Grünbaum, B. (1983). Tilings, patterns, fabrics and related topics in discrete geometry. Jber. Dtsch. Math.-Verein. 85, 1–32.
Grünbaum, B. & Shephard, G. C. (1981). A hierarchy of classification methods for patterns. Z. Kristallogr. 154, 163–187.
Koch, E. (1973). Wirkungsbereichspolyeder und Wirkungsbereichsteilungen zu kubischen Gitterkomplexen mit weniger als drei Freiheitsgraden. Z. Kristallogr. 138, 196–215.
Koch, E. (1984). A geometrical classification of cubic point configurations. Z. Kristallogr. 166, 23–52.
Koch, E. & Fischer, W. (1978). Types of sphere packings for crystallographic point groups, rod groups and layer groups. Z. Kristallogr. 148, 107–152.
Koch, E. & Hellner, E. (1971). Die Pattersonkomplexe der Gitterkomplexe. Z. Kristallogr. 133, 242–259.
Naor, P. (1958). Linear dependence of lattice sums. Z. Kristallogr. 110, 112–126.
Smirnova, N. L. (1962). Possible values of the x coordinates in single-parameter lattice complexes of the cubic system. Sov. Phys. Crystallogr. 7, 5–8.

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