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On a space-fillingpolyhedron of Aste et al.M. O'KeeffePublished online: 14 Nov 2010.

To cite this article: M. O'Keeffe (1997) On a space-filling polyhedronof Aste et al., Philosophical Magazine Letters, 76:6, 423-426, DOI:10.1080/095008397178850

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PHILOSOPHICAL MAGAZINE LETTERS, 1997, VOL. 76, NO. 6, 423± 426

On a space-® lling polyhedron of Aste et al.

By M. O’Keeffe

Department of Chemistry, Arizona State University, Tempe,Arizona 85287, USA

[Received 1 July 1997 and accepted 12 August 1997]

AbstractA 16-face polyhedron described by Aste, Boose and Rivier (1996 Phys. Rev. E,

53), is shown to ® ll space with four polyhedra meeting at each vertex and threemeeting at each edge. Coordinates of the vertices in the resulting periodicstructure are given for the conformation with equal edges and maximum volume.

There is considerable interest in packings of simple polyhedra (i.e. those in whichthree faces meet at each vertex) to form networks in which four polyhedra meet ateach vertex and three meet at each edge. In chemistry, such structures serve as theframeworks of many crystal structures such as those of clathrate hydrates and ofzeolites and other framework aluminosilicates (for example O’Kee� e and Hyde(1996)) and they are also of interest as possible structures of foams or froths. Of cen-tral interest in these connections are space ® llings by congruent polyhedra(`monotilings’ ). A well known example of the latter is the packing of truncated octa-hedra (tetrakaidecahedra) to form the structure known to chemists as the sodaliteframework and to the foam community as the Kelvin structure (Weaire 1997). Otherexamples have been adduced by Williams (1968) and by Weaire and Phelan (1994). Allthese monotiling polyhedra have 14 faces and, as a consequence of Euler’ s theoremapplied to simple polyhedra, the average number of sides per face is k nl = 36

7 .For simple polyhedron packings that form the basis of known crystal structures,

k nl £ 367 and O’Kee� e and Hyde (1996) speculated that this might be the maximum

possible, but (to invert a well known aphorism) this is an example of somethingwhich (some) chemists believe to be true, but which mathematicians know to be false.

In a study of certain types of froth, Aste et al. (ABR) (1996) derived some of theknown 14-face monotiling polyhedra and also described the topology of a 16-facepolyhedron of which they wrote `As far as we know, this cell is a [new] monotileEuclidean space ® ller.’ The cell has k nl = 21

4 = 5.25 (i.e greater than 367 = 5.14. . .); so

it is of interest to determine whether indeed it does tile space and the nature of thetiling. Accordingly such a structure is described.

The nets occurring in crystal structures have (at least approximately) equal edgelengths which correspond to the shortest distances between vertices, so that all otherinter-vertex distances are larger than the edge lengths. O’Kee� e and Brese (1992)have referred to nets which have this property, when the volume is maximizedsubject to constraint of equal edge length, as r̀ealizable’ (it appears that all netsof interest in crystal chemistry are realizable in this sense). The volume maximizationprocedure assures that the conformation is that of highest symmetry and I haveapplied it to a periodic packing of the ABR tiles. The result shows that the net isindeed realizable, and the maximum volume form is described next.

0950± 0839/97 $12 ´00 Ñ 1997 Taylor & Francis Ltd.

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The structure is tetragonal with two polyhedra per unit cell. The symmetry isP42 /mcm with cell parameters a = 2.5479d, c = 3.7369d, where d is the length of theedges of the 4-connected net. The vertices are characterized by a vertex symbol(O’Kee� e 1991) in which the six numbers specify the size of the rings (polyhedronfaces in this instance) meeting at the vertex, with pairs of opposite angles appearingsequentially. They are of three kinds: two 4.4.4.4.6.6 at 6 (0,0, 1

4) , four 4.4.6.6.8.8at 6 (0, 1

2 , 14 ; 1

2 ,0, 14) and eight 4.6.4.6.6.8 at 6 (x,x, 6 z; x,x, 1

2 6 z) with x = 14 and

z = 0.3662. The structure has four variable parameters a, c, x and z and threeindependent edge lengths; so the volume can indeed be varied with constant edgelength, and the parameters given are for maximum volume. The shortest distancebetween vertices not joined by an edge is 1.274d.

The polyhedron is illustrated in ® gure 1. As shown in the ® gure it may be derivedfrom a face-sharing pair of elongated rhombic dodecahedra by eliminating the

424 M. O’Kee� e

(a)

(b)

(c)

Figure 1. The ABR polyhedron: (a) with transparent faces; (b) with opaque faces; (c) twoelongated rhombic dodecahedra sharing a hexagonal face.

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shared face and two vertices. The elongated rhombic dodecahedron is well known asthe Voronoi cell of the bct lattice with c /a > 21/2 and is one of the ® ve parallelohedraof Federov. The corresponding pattern of ® lling space is illustrated in ® gure 2. Notethat to ® ll space the octagonal faces would have to be curved rather than made up ofplanar facets as shown in the ® gure. The angles at the vertices vary from 79ë to 129 ë .The space ® lling of elongated dodecahedra occurs conspicuously in the crystal struc-ture named for ThCr2Si2 with Cr and Si at the verticec and Th centring the poly-hedra. This is in fact the most populous of all ternary crystal structure types (Pearson1985). In the structure corresponding to packed ABR polyhedra, a quarter of the Crpositions are empty; so the composition of the hypothetical structure is Th2Cr3Si4. Ileave questions concerning the stability and likely evolution of a foam of suchbubbles to those better quali® ed.

An obvious feature of the polyhedron shown in ® gure 1 is that the octagonalfaces are very nonplanar, and the whole polyhedron is far from convex. Figure 3shows a di� erent conformation of the same polyhedron with ¯ at faces; now somedistances between non-adjacent vertices are equal to the edge length. Note too thatsome dihedral angles are necessarily 180 ë or greater. In all examples of polyhedronpackings of which I am aware, in which k nl > 36

7 and in which the polyhedra have

Space-® lling polyhedron of Aste et al. 425

Figure 2. The packing of ABR polyhedra. Note that to ® ll space completely the octagonalfaces (darker shading) would have to be curved.

Figure 3. The ABR polyhedron with planar faces.

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equal edges, some of the dihedral angles are 180 ë or greater. This makes themunattractive as candidates for the structure of bubbles in foams (although bubblesare not restricted to having equal edge length). In any event it does appear that itmight be fruitful to enquire into the restrictions (if any) on the nature of the poly-hedra in packings of simple polyhedra for which n > 36

7 .The shell structure of foams is also of interest (Aste et al. 1996). This is char-

acterized by the number of polyhedra in successive concentric shells about a centralpolyhedron. For the Kelvin structure the number in the kth shell is nk = 12k2 + 2(Marvin 1939). The corresponding sequence for the ABR structure has beendetermined by direct enumeration (O’Kee� e 1995) and is n1 = 16, n2 = 70 andnk = 20k2 - 4k + 2 for k > 2.

ACKNOWLEDGEMENTS

I am grateful to J.-F. Sadoc and N. Rivier for an opportunity to attend theNATO school on foams at Cargese, Corsica, in May 1997. Conversations therewith T. Aste, who called my attention to this problem, and with J. Sullivan, whodemonstrated to me that polyhedron packings with k nl > 36

7 are indeed possible, areacknowledged. This work was supported by the US National Science Foundationunder grant No. DMR 94 24445.

REFERENCES

Aste, T., Booseí, D., and Rivier, N., 1996, Phys. Rev. E, 53, 6181.Marvin, J. W., 1939, Science, 83, 188.O’ Keeffe, M., 1991, Z. Kristallogr., 196, 21; 1995, ibid., 210, 905.O’ Keeffe, M., and Brese, N. E., 1992, Acta Crystallogr. B, 48, 152.O’ Keeffe, M., and Hyde, B. G., 1996, Crystal Structures I: Patterns and Symmetry

(Washington, DC: Mineralogical Society of America).Pearson, W. B., 1985, J. solid-st. Chem., 56, 278.Weaire, D. (editor), 1997, The Kelvin Problem (London: Taylor & Francis).Weaire, D., and Phelan, R., 1994, Phil. Mag. L ett. , 70, 345.Williams, R. E., 1968, Science, 161, 276.

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