Interesting packings of spheres in cubic space groups. A case of an unobvious arrangement of space-filling rhombohedra

Close packing of spheres has long intrigued people. The (unpublished) discovery of the densest possible packing with (later established as) cubic symmetry Fm3̅m is attributed to Thomas Harriot (Kahr, 2011), although the credit is usually given to Johannes Kepler, who published this concept in his famous booklet De Nive Sexangula (Kepler, 1611, 1966), considered the first modern book on crystallography. The equally dense canonical hexagonal packing and another hexagonal example were discovered by William Barlow (1883). Later, it was realised that there are, in fact, infinitely many such hexagonal packings, with elegant analogies to bracelet topology and integer number partition in number theory, as summarised by Jaśkolski, Naskręcki & Dauter (2025). Famously, Linus Pauling (1932, 1960) was also interested in this problem, seeing its relation to the packing of simple crystal structures. Sphere packing is indeed closely related to structural science and technology (powder metallurgy), to logistics (container packing), and even to unexpected areas such as data communication (error-correcting codes).     

Now, regular sphere packing means that the arrangement of symmetry equivalent spheres in 3D is fully connected, meaning that for any pair of spheres, there is always a path leading through the sphere kissing points. With such a generous definition, the area offers a vast number of possibilities.

In the course of our comprehensive analysis of the close-packing problem across all cubic space groups, which builds on the fundamental work of Werner Fischer (1973), we came across several intriguing, non-intuitive 3D space tessellations. We intend to present them as short essays in the IUCr Newsletter. One such case, found in the Pa3̅ space group, revealed an unusual tessellation of the 3D space with rhombohedra.

It is well known that identical, non-cubic rhombohedra can completely fill the 3D space if arranged in a parallel fashion as unit cells in the space groups R3, R3̅, R32, R3m, R3c, R3̅m, R3̅c, for any ratio of the a/c cell dimensions (in the hexagonal setting) and for any angle α between the edges of the rhombic faces of these polyhedra. In each case (except the square), the rhombic face can be used to build either a prolate or oblate rhombohedron, i.e. a version with a longer or shorter body diagonal along the threefold axis, but a given rhombohedral space tessellation with the unit cells will always use a rhombohedron of one type only. (A side note: as an exception, only one special shape of the rhombohedron, the cube, forms a parallel arrangement with cubic symmetry in the Pm3̅m space group.)

Since the rhombohedra have all edges of equal length, placing spheres of diameter equal to the edge length at the corners of such a rhombohedral tessellation will form proper packing, because there will always be a path between any pair of spheres through a number of other touching spheres. The underlying rhombohedra can then be treated as a representation of the voids between the touching spheres. In particular, the centres of those polyhedra coincide with the centres of the packing voids.

Werner Fischer (1973, 1974), in his cataloguing of close-packed arrangements in the cubic space groups, presented his solutions in tabulated form as sphere coordinates. (N.B. There are several errors in his presentation, as well as omissions, to be discussed in our full paper.) However, he never went beyond those listings, i.e. did not give any interpretations of the topological aspects of those solutions, such as interstitial void geometry or coordination. What we are presenting here is a partial extension of his tabulations.

In the list of Fischer’s packings, there is one example (1973, Table 3, p. 135, case 7/3/c1) where the inter-sphere voids would be of two complementary shapes, corresponding to a prolate and oblate rhombohedron. This pair of rhombohedra must, of course, fill the 3D space without gaps, but in an arrangement that is different from the usual parallel fashion characteristic of trigonal symmetry. Both such polyhedra have identical rhombic faces with diagonals equal to 2√3/3 (≈1.155) and 2√6/3 (≈1.633) times the edge length and with the “tetrahedral” values of α and 180° - α angles of 109.5° and 70.5° (Fig. 2).

   

The length of the body diagonal of the oblate rhombohedron is equal to the length of its edge. The body diagonals of both rhombohedra are the directions of threefold symmetry (Fig. 1).

As mentioned above, the discussed rhombohedra correspond to close packing of spheres located at the Wyckoff position c of the Pa3̅ space group and, therefore, the space tessellation by a pair of these (properly assembled) rhombohedra also has the cubic space group Pa3̅ symmetry, as shown in Fig. 3.

 

In the Pa3̅ unit cell the corners of the polyhedra lie at the threefold axis at position x=y=z=(√5-1)/8=0.1545 and at all other symmetry-equivalent sites of the Wyckoff c position (x, x, x) (Int. Tables for Crystallography, Vol. A, 2016). The centres of the prolate rhombohedra lie at the Wyckoff b positions (½,½,½ and equivalent), and the centres of the oblate rhombohedra occupy the a positions (0,0,0 and equivalent), both with 3̅ point symmetry.

All corners of all rhombohedra are surrounded by 7 other corners at the same distance. The coordination polyhedron around the individual sphere in this packing is shown in Fig. 4.

 

In the next issue, we intend to present the case of the generation of the loosest Heesch-Laves (1933) packing of equal spheres and its crystallographic symmetry. Stay tuned.

References

Barlow, W. (1883). “Probable Nature of the Internal Symmetry of Crystals.” Nature 29, 186-188.

International Tables for Crystallography, Vol. A (2016). Space-group Symmetry, 6^th ed. Aroyo, M.I. ed. Chichester, Wiley.

Fischer, W. (1973). „Existenzbedingungen homogener Kugelpackungen zu kubischen Gitterkomplexen mit weniger als drei Freiheitsgraden.“ Z. Krist. 138, 129-146.

Fischer, W. (1974). „Existenzbedingungen homogener Kugelpackungen zu kubischen Gitterkomplexen mit drei Freiheitsgraden.“ Z. Krist. 140, 50-74.

Heesch, H. & Laves, F. (1933). “Uber dunne Kugelpackungen.” Z. Krist. 85, 443-453.

Jaskolski, M., Naskrecki, B. & Dauter, Z. (2025). „Periodic arrangements of closely packed spheres.” ChemTexts 11:2.

Kahr, B. (2011). “Et Tu, Crystallographer? Murder Charges Against Close-Packing Pioneers Evaluated.” Cryst. Growth Des. 11, 4-11.

Kepler, J. (1611). Strena Seu De Nive Sexangula. G. Tampach, Francofurti ad Moenum.

Kepler, J. (1966). The Six-Cornered Snowflake. Clarendon Press, Oxford.

Pauling, L. (1932). “The packing of spheres.” Chem. Bull. Chicago 19, 35-38.

Pauling, L. (1960). The Nature of the Chemical Bond. 3^rd ed. Cornell Univ. Press. Ithaca.