Crystallography Reports, Vol. 47, No. 5, 2002, pp. 709–719. Translated from Kristallografiya, Vol. 47, No. 5, 2002, pp. 775–784. Original Russian Text Copyright © 2002 by Talis. CRYSTAL SYMMETRY Generalized Crystallography of Diamond-Like Structures: II. Diamond Packing in the Space of a Three-Dimensional Sphere, Subconfiguration of Finite Projective Planes, and Generating Clusters of Diamond-Like Structures A. L. Talis Institute of Synthesis of Mineral Raw Materials, Aleksandrov, Vladimir oblast, Russia e-mail: ofi@vniisims.elcom.ru Received April 24, 2001 Abstract—An enantiomorphous 240-vertex “diamond structure” is considered in the space of a three-dimen- sional sphere S3, whose highly symmetric clusters determined by the subconfigurations of finite projective planes PG(2, q), q = 2, 3, 4 are the specific clusters of diamond-like structures. The classification of the gener- ating clusters forming diamond-like structures is introduced. It is shown that the symmetry of the configuration, in which the configuration setting the generating clusters is embedded, determines the symmetry of diamond- like structures. The sequence of diamond-like structures (from a diamond to a BC8 structure) is also considered. On an example of the construction of PG(2, 3), it is shown with the aid of the summation and multiplication tables of the Galois field GF(3) that the generalized crystallography of diamond-like structures provides more possibilities than classical crystallography because of the transition from groups to algebraic constructions in which at least two operations are defined. © 2002 MAIK “Nauka/Interperiodica”. INTRODUCTION the diamond-like structures themselves are constructed from a certain set of generating clusters in a way similar Earlier [1], it was shown that the adequate mapping to the construction of a three-dimensional Penrose divi- of the symmetry of diamond-like structures requires a sion from four types of tetrahedra [7]. change of the Euclidean basis of the structural crystal- lography to a more general basis of algebraic (projec- The most highly symmetric diamond-like structure tive) geometry. In particular, the incidence graphs of is such a combination of two three-dimensional lattices specific subconfigurations of finite projective planes that can be implemented in the Euclidean space E3 as a PG, q = 2, 3, 4 turned out to be isomorphous to the diamond and in the space of a three-dimensional sphere graphs of the specific clusters of a diamond-like struc- S3 as an “enantiomorphous diamond” or an irregular ture. These graphs are mapped onto themselves by the {240} polytope [4, 5, 8]. This shows the importance of groups of projective geometry and contain the orthogo- those specific clusters of the {240} polytope in the gen- nal groups of the classical crystallography as their sub- eralized crystallography of diamond-like structures that groups. are defined by the corresponding subconfigurations The present article is the second of a series dedi- PG(2, q) considered in detail in the present study. cated to the construction of the generalized crystallog- raphy of diamond-like structures as a particular struc- The frequent detailed citation of [1] allows the tural application of algebraic geometry, which includes author to use the definitions introduced in [1] and the as the limiting case the respective sections of the clas- results obtained there. For clarity, the author often sac- sical crystallography [2, 3]. For diamond-like struc- rifices mathematical rigor and uses numerous illustra- tures, the penta-, hexa-, and heptacycles are the most tions to be able to compare directly the mathematical realistic [4–6]. Therefore, the present study is dedicated constructions with the clusters they determine. The mainly to the determination of the symmetrically pos- results obtained make the basis for constructing the sys- sible types of generating clusters of diamond-like struc- tem of generating clusters, which allows the author to tures containing only these cycles. In the general case, derive diamond-like structures and determine all the the specific clusters of a diamond-like structure are the symmetrically possible structural phase transitions combinations of various generating clusters, whereas between these structures. 1063-7745/02/4705-0709 $22.00 © 2002 MAIK “Nauka/Interperiodica” 710 TALIS X3 (a) (b) (c) Ð1,0,τ 1,0,τ X1 Ð1,0,Ðτ 1,0,Ðτ (d) (e) (f) X3 (g) 0,0,x(0) τ, , (1) 3 (1) Ð x2 x 3 2,0,x 3 3 22 11 14 30 25 19 6 8 7 17 27 6 28 16 R1 5 9 5 X2 20 24 1 4 13 3 12 2 R2 23 21 X1 4 10 2 15 26 7 18 29 ÐÐ 2 R3 53m 1 Fig. 1. Mapping of substructures of polytopes in E3 [5, 10, 12]: (a) {3, 5} icosahedron whose three twofold axes coincide with the X1-, X2-, and X3 axes of the Cartesian coordinate system (four hatched triangles and the center of the icosahedron belong to four tetrahedra whose centers form a tetrahedron); (b) {4, 3} cube and its projection onto the plane represented in the form of a Schlegel diagram (six-edge Petrie’s polygon of the cube and its Schlegel diagrams are shown by solid lines); (c) the projection of the {4, 3, 3} polytope in E3; solid lines show its eight-edge Petrie’s polygon; (d) the icosahedron and the stereographic projection of the 53m group, whose fundamental domain is separated by the reflection planes R1, R2, and R3 shown by solid lines (the fivefold axis of the icosahedron coincides with the X3-axis of the Cartesian coordinate system; the sections of the icosahedron by the plane normal to ()0 X3-axis are the (0, 0, x3 ) point, the hatched pentagon, etc.; a 10-edge Petrie’s polygon of the {3, 5} icosahedron is shown by solid lines); (e) the rod of icosahedra sharing the fivefold axis (the part of the helix determined by the 101 screw axis of the {3, 3, 5} polytope is shown by a solid line); (f) Bernal’s chain of tetrahedra; the 30/11 irrational screw helix corresponding to Petrie’s polygon of the {3, 3, 5} polytope is shown by solid arrows; (g) the starlike polygon formed as a result of the orthogonal projection of 30 vertices of the {3, 3, 5} polytope along the 30/11 axis. A 240-VERTEX “DIAMOND” STRUCTURE (cos2(π/h) = cos2(π/p) + cos2(π/q)) in which any three IN THE SPACE OF A THREE-DIMENSIONAL successive sides do not belong to one face. Thus, for a SPHERE S3—AN IRREGULAR {240} POLYTOPE {4, 3} cube, h = 6 (Fig. 1b). For a {3, 5} icosahedron, Petrie’s polygon consists of h = 10 “side” edges of a Three-dimensional Platonic solids can be general- ized to projective Platonic solids—finite projective pentagonal antiprism (Fig. 1d). For a {p, q, r} polytope, planes PG(2, q). Taking into account the metric rela- Petrie’s polygon is a set of edges in which any succes- tionships, this generalization requires the transition sive three (and not four) edges belong to the Petrie’s polygon of the {p, q} cell [10]. For example, the Pet- from the space E3 to the space En, n > 3. At n = 4, the rie’s polygon of a four-dimensional cube (the {4, 3, 3} Platonic solids in the Schläfli notation {p, q} (Figs. 1a, 1b, 1d) are extended to the four-dimensional Platonic polytope) is an octacycle, any three successive edges of solids—{p, q, r} polytopes built by the {p, q} cells in which belong to one cubic {4, 3} cell of this polytope such a way that each face {p} is shared by two cells and (Fig. 1c). each edge, by r cells. The location of the cells at the ver- The division of E3 into regular tetrahedra is impos- tex {p, q, r} corresponds to the location of the faces of the sible because the dihedral angle of the regular tetrahe- {q, r}-vertex polyhedron of the polytope (Fig. 1c) [9–12]. dron equals 70.53¡ < 72¡ = 360¡/5. The “angular defi- The most important characteristic of the n-dimen- cit” of the connected tetrahedra can be avoided if one sional polyhedron is its Petrie polygon; for the {p, q}- “sacrifices” their regularity by connecting 20 tetrahedra polyhedron, it is a “zig-zag” consisting of h edges into an icosahedron (Fig. 1a), which can be made only CRYSTALLOGRAPHY REPORTS Vol. 47 No. 5 2002 GENERALIZED CRYSTALLOGRAPHY OF DIAMOND-LIKE STRUCTURES: II. 711 by using two types of edges with the length ratio the number of edges of the Petrie’s polygon of the ~0.951 in the tetrahedra. In this case, 12 vertices of an {3, 3, 5} polytope [13, 14]. Code (3) determines the {3, 5} icosahedron provide the regular triangulation of presence in the polytope {3, 3, 5} of the axes which in the sphere S2, which is mapped onto itself by the E3 coincide with the axes of rotation by 2π/n, n = 2, 3, orthogonal group 53m of order 120 set by the follow- 5 (Fig. 1d), the conventional (rational) screw axes n1, ing defining relationships for the generating elements n = 6, 10 (Fig. 1e), and the irrational screw axis 30/11, (code): which provides the rotation by about 2π11/30 around 11/30 with the subsequent translation along this axis 2 ()3 ()5 ()2 Ri ===R1 R2 R2 R3 R1 R3 (Fig. 1g). The 30/11 axis reflects the symmetry of a (1) 30-edge Bernal helix composed by tetrahedra (Fig. 1f) ()10 = R1 R2 R3 = 1, and corresponding to the Petrie’s polygon of the {3, 3, 5} polytope [9, 10, 15]. where Ri, i = 1, 2, 3 are the generating reflection planes that single out the fundamental domain of the group In the {3, 3, 5} polytope, one can select 120 tetrahe- dra whose centers form the congruent ϕ{3, 3, 5} poly- 53m ; R1R2, R2R3, R1R3 are the rotations by the angles π π π tope, where ϕ is the enantiomorphous rotation in E4 and of 2 /3, 2 /5, and , and R1R2R3 = 5 is the Coxeter ele- ∋ ϕ ∉ ment whose degree h = 10 coincides with the number of O(4) [3, 3, 5]; in this case, in each icosahedron edges in the Petrie’s polygon of an icosahedron from the {3, 3, 5} polytope, four tetrahedra are centered (Fig.