International
Tables for
Crystallography
Volume C
Mathematical, physical and chemical tables
Edited by E. Prince

International Tables for Crystallography (2006). Vol. C, ch. 9.1, p. 751

## Section 9.1.2. Packings of ellipses and ellipsoids

E. Kocha and W. Fischera

aInstitut für Mineralogie, Petrologie und Kristallographie, Universität Marburg, Hans-Meerwein-Strasse, D-35032 Marburg, Germany

### 9.1.2. Packings of ellipses and ellipsoids

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The problem of deriving packings of ellipses in two-dimensional space or of ellipsoids in three-dimensional space may be regarded as a generalization of the problem of deriving circle packings and sphere packings. It is much more complicated, however, because a circle or sphere is fully determined by its centre and its radius, whereas the knowledge of the centre, the lengths of the two semiaxes, and the direction of one of them is needed to construct an ellipse. For an ellipsoid, the knowledge of its centre, the length of its three semiaxes, and the directions of two of them is necessary. Accordingly, the point configuration corresponding to the ellipsoid centres does not define the ellipsoid packing and not even its type.

Nowacki (1948) derived 54 homogeneous essentially different packings of ellipses'. In contrast to the definition of types of sphere (circle) packings (Section 9.1.1), Nowacki distinguished between similar packings with different plane-group symmetry, i.e. between packings that may differ in the orientation of their ellipses. Under an equivalent classification, Grünbaum & Shephard (1987) derived 57 different cases of ellipse packings, thus correcting and completing Nowacki's list. Each of these 57 cases corresponds uniquely to one of the 11 types of circle packings if one takes into account only the contact relations between ellipses and circles. In eight cases, each ellipse has six contacts. Two of these cases can be derived from the densest packing of circles by affine transformations and, therefore, have the same density, namely ρ = 0.9069, irrespective of the shape of the ellipses (Matsumoto & Nowacki, 1966). Presumably for the other six cases this density can only be reached (but not exceeded) if the ellipses become circles. A corresponding proof is in progress (Matsumoto, 1968; Tanemura & Matsumoto, 1992; Matsumoto & Tanemura, 1995).

Very little systematic work seems to be carried out on homogeneous or heterogeneous packings of ellipsoids. Matsumoto & Nowacki (1966) derived packings of ellipsoids with contact numbers 12 and high densities by affine deformation of cubic and hexagonal closest packings of spheres. They postulate (without proof) the following: Densest packings of ellipsoids have the same contact number and density as closest packings of spheres and can be derived always from closest sphere packings by affine transformations. If this assumption is true, densest packings of ellipsoids would necessarily consist of parallel ellipsoids only.

Packings of ellipsoids seemed to be useful for the interpretation of the arrangements of organic molecules in crystals. The studies of Kitaigorodsky (1946, 1961, 1973), however, showed that molecular crystals may rather be regarded as dense packings of molecules with irregular shape.

Heterogeneous packings of ellipsoids may possibly be adequate for the geometrical interpretation of some intermetallic compounds like cubic MgCu2 (cf. Subsection 9.1.1.4) or Cr3Si. The ellipsoids enable the use of different atomic radii' with respect to neighbouring atoms of the same kind or of different kinds. In MgCu2, for example, the magnesium atoms have cubic site symmetry and therefore can only be represented by spheres. The Cu atoms [16(d)] with site symmetry , however, may be represented by flattened ellipsoids of revolution.

### References

Grünbaum, B. & Shephard, G. C. (1987). Tilings and patterns. New York: Freeman.
Kitaigorodsky, A. I. (1946). Arrangement of molecules in organic crystals. Thesis, Moscow, Russia. [In Russian.]
Kitaigorodsky, A. I. (1961). Organic chemical crystallography. New York: Consultants Bureau. [Russian text published by Press of the Academy of Sciences of the USSR, Moscow, 1955.]
Kitaigorodsky, A. I. (1973). Molecular crystals and molecules. New York/London: Academic Press.
Matsumoto, T. (1968). Proof that the pgg packing of ellipses has never the maximum density. Z. Kristallogr. 126, 170–174.
Matsumoto, T. & Nowacki, W. (1966). On densest packings of ellipsoids. Z. Kristallogr. 123, 401–421.
Matsumoto, T. & Tanemura, M. (1995). Density of the p3 packing of ellipses. Z. Kristallogr. 210, 585–596.
Nowacki, W. (1948). Symmetrie und physikalisch–chemische Eigenschaften kristallisierter Verbindungen. V. Über Ellipsen­packungen in der Kristallebene. Schweiz. Mineral. Petrogr. Mitt. 28, 502–508.
Tanemura, M. & Matsumoto, T. (1992). On the density of the p31m packing of ellipses. Z. Kristallogr. 198, 89–99.