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Crystallography Reports, Vol. 47, No. 4, 2002, pp. 527–536. Translated from Kristallografiya, Vol. 47, No. 4, 2002, pp. 583–593. Original Russian Text Copyright © 2002 by Talis. THEORY OF CRYSTAL STRUCTURE Generalized Crystallography of Diamond-Like Structures. 1. Finite Projective Planes and Specific Clusters of Diamond-Like Structures Determined by These Planes A. L. Talis Institute of Synthesis of Mineral Raw Materials, Aleksandrov, Moscow oblast, Russia e-mail: ofi@vniisims.elcom.ru Received March 20, 2001; in final form, March 6, 2002 Abstract—It is shown that the adequate mapping of the symmetry of determined crystalline and noncrystalline diamond-like structures requires a change in the Euclidean basis of the structural crystallography to a more gen- eral basis of projective (algebraic) geometry. It is established that the incidence graphs of specific subconfigu- rations of a finite projective plane PG(2, q) with q = 2, 3, 4 are isomorhic to the graphs of specific clusters of diamond-like structures. The clusters thus determined are invariant with respect to the groups of projective geometry that have orthogonal groups as subgroups and, in fact, are set by the subtables of the incidence tables PG(2, q) with q = 2, 3, 4. As an example, the a priori derivation of clusters, which are not traditional crystallo- graphic objects, is considered, namely, the parallelohedron of the diamond structure, the combination of two Bernal polyhedra Z8 and clusters containing icosahedra. © 2002 MAIK “Nauka/Interperiodica”. INTRODUCTION spaces, S3 and H3 [5, 6], the theory of stratified spaces [7], and the lattices of roots [8]. However, up to now, no The importance of determined diamond-like struc- mathematical apparatus for adequate mapping of the tures for fundamental crystallography and their appli- symmetry of the determined (not only crystal) dia- cation can hardly be overestimated. Diamond-like mond-like structures has been developed within all structures are formed in high-pressure crystalline the mentioned extensions of classical crystallogra- phases, crystals with linear defects, thin films, nanoma- phy, either because the mathematical basis was terials, etc., whose structure types are determined by a developed insufficiently or, on the contrary, because tetrahedrally coordinated graph consisting of a set of the most complicated mathematical apparatus was vertices and edges. For several decades, the a priori confirmed only by a few rather abstract two-dimen- derivation of the structure types of crystals was based sional examples. on the empirical and intuitive construction of such graphs, and only in 1999, the efforts of mathemati- We believe that the necessary mathematical appara- cians (a group headed by A. Dress) and crystallogra- tus for the adequate mapping of the symmetry of a dia- phers (A. Mackay) were crowned with the determina- mond-like structure should be similar to that of the tion of all the crystalline tetrahedrally coordinated space groups and represented in the form of a clear but combinatorially and topologically different graphs of mathematically rigorous basis. The results obtained in the divisions of the three-dimensional Euclidean space our earlier studies [2, 9–11] show that all these require- E3 into convex polyhedra with vertices of one, two, or ments are met by the projective (in a wider sense, alge- three different types [1]. In fact, the mathematical appa- braic) geometry whose joint scheme includes Euclidean, ratus used in [1] is based on the algebraic constructions, Lobachevski, and Riemannian geometries and deals which are more general than discrete groups of motion with the objects whose consideration in E3 uniquely in E3. Thus, the adequate mapping of the symmetry of determines diamond-like structures [12]. a crystalline structure (to say nothing of the determined The aim of the present study is to construct the gen- noncrystalline structures) cannot be made in terms of eralized crystallography of diamond-like structures as a the classical crystallography limited by space groups specific structural application of algebraic geometry, defined in E3. which includes, as the limiting cases, also the corre- Lifting the main restrictions of classical crystallog- sponding sections of classical crystallography. raphy requires a transition to En at n > 3 and to non- The construction of the generalized crystallography Euclidean geometries. The most important examples of diamond-like structures allows one to derive a priori here are the studies of quasicrystals [2–4], regular divi- defined diamond-like structures and determine possible sions of three-dimensional Riemann and Lobachevski phase transitions in these structures. The criterion of the 1063-7745/02/4704-0527 $22.00 © 2002 MAIK “Nauka/Interperiodica” 528 TALIS (a) (b) N P2 0 z = 0 0 P' P1 z = 1 P z = 1 P Fig. 1. Projection of a sphere onto Euclidean and projective planes. (a) The sphere S2 is uniquely projected from the North pole N onto the plane E2 and (b) a pair of points lying on the opposite ends of the diameter of the sphere S2 are projected from the center of the sphere onto one point of the projective plane PG(2, K). correctness of this generalized crystallography is the tures are uniquely defined by certain mathematical con- independent reproduction of the results obtained in [1, structions represented in tabulated form. 3–6, 13–17] within the framework of this generalized theory. The inefficiency of the synthesis and the study of crystals without the use of classical crystallography DEFINITIONS AND TERMINOLOGY leads to the conclusion that the application of the gen- A projective plane PG(2, K) consists of a Euclidean eralized crystallography of diamond-like structures in plane E2 = EG(2, K) and one added point at the infinity these studies and the synthesis of the determined dia- for each pencil of straight lines parallel in E2. The mond-like structures would facilitate the efforts of model of PG(2, K) is shown in Fig 1b. chemists, physicists, and scientists working in materi- The straight line in the general position in E3 pass- als and other sciences in their labor-consuming empiri- ing through the center of a sphere does not lie in the cal and intuitive structural constructions. In virtue of plane z = 0, and, thus, intersects the plane z = 1 only at the tetrahedral coordination the characteristic of zeo- one point. The straight lines lying in the plane z = 0 do lites, silicates, gas hydrates, fullerites, etc., certain sec- not intersect the plane z = 1, and, therefore, they corre- tions of the generalized crystallography of diamond- spond not to the points from E2, but to the pencils of like structures can also be successfully used in the stud- parallel straight lines from E2. The structure containing ies of these materials [17]. For mathematicians, the n points P1, P2, …, Pn and n straight lines l1, l2, …, ln in generalized crystallography of diamond-like structures which each point is intersected by the q + 1 “straight can be an interesting example of mathematical applica- line” (n = q2 + q + 1) consisting of q + 1 points is called tions. a finite projective plane PG(2, q) of order q (Fig. 2a). The present article is of an introductory nature and The plane PG(2, q) is uniquely defined by its incidence × considers examples that clearly show that, at least, for table in the form of a square n n table in which the col- diamond-like structures, the Euclidean basis of the umns P1, P2, …, Pn are called the points, and the rows structural crystallography should be substituted by a l1, l2, …, ln, the straight lines. The incidence of the point more general basis of projective (algebraic) geometry. Pj and the straight line li in this table are defined by We consider the examples of clusters of diamond-like placing a black circle into the cell ij; the empty cell sig- structures whose symmetry can be adequately mapped nifies the absence of incidence (Fig. 2). For PG(2, q), only at the level of the groups of projective geometry the incidence table (IT(q)) can be constructed using the including the orthogonal groups as their subgroups. In summation and multiplication tables of the Galois field many instances, mathematical rigor was often sacri- GF(q), of order q [8, 18, 19]. ficed for the sake of a clearer representation, and pro- The set of m points and n straight lines (planes) in vided the reader with numerous illustrations that which f straight lines (planes) pass through each point allowed a direct comparison of the constructions of the and d points lie on each straight line (plane) is called a projective geometry with the clusters of the diamond- configuration. The configuration is denoted as (mf , nd); like structure uniquely defined by these structures. In if m = n, the configuration is self-dual and is denoted as other words, to show the great possibilities of the gen- nd (Figs. 2a and 3a). The configuration is defined by its eralized crystallography of diamond-like structures one incidence table (IT(mf , nd)) consisting of m rows and n should not necessarily read the whole article; it is suffi- columns and containing m · f = n · d incidence signs. cient to get acquainted with the figures showing that the The configuration can be implemented in various (finite real and hypothetical clusters of diamond-like struc- or infinite) spaces; if it is implemented on a certain pro- CRYSTALLOGRAPHY REPORTS Vol. 47 No. 4 2002 GENERALIZED CRYSTALLOGRAPHY OF DIAMOND-LIKE STRUCTURES 529 (a) (b) (c) P l (∞)(1) (0) 2 2 l1 P1 P2 P3 P4 P5 P6 P7 l5 P5 P l1 P4 l4 3 l2 P2 l2 P7 l7 l7 l5 (1, 1) P1 l3 (0, 1) (1, 0) l4 l6 P6 l6 P5 P7 l5 l3 P3 l6 l3 l7 (0, 0) l1 P1 P6 P4 l4 (e) (d) 4 1 l3 l2 l7 2 2 P1 P5 1 3 3 l6 l1 l5 l3 00 3 5 4 P3 P2 P6 6 5 6 4 7 l l l 6 1 2 2 7 7 4 6 4 1 P5 P4 P7 P3 5 5 3 l3 l2 P6 P1 P5 6 Fig.

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