Crystallography Reports, Vol. 49, No. 4, 2004, pp. 527–536. From Kristallografiya, Vol. 49, No. 4, 2004, pp. 599–609. Original English Text Copyright © 2004 by Watanabe, Soma, Ito. CRYSTALLOGRAPHIC SYMMETRY A Quasi-Lattice Based on a Cuboctahedron. 1. Projection Method1 Y. Watanabe*, T. Soma**, and M. Ito*** * Department of Information Systems, Teikyo Heisei University, 2289 Uruido, Ichihara, Chiba, 290-0193 Japan e-mail: [email protected] ** Computer and Information Division, Riken, Wako-shi, Saitama, 351-0198 Japan *** Faculty of Engineering, Gunma University, Aramaki, Gunma, 371-8510 Japan Received September 1, 2003 Abstract—A three-dimensional (3D) quasi-lattice (QL) based on a cuboctahedron is obtained by the projection from a seven-dimensional hypercubic base lattice space to a 3D tile-space. The projection is defined by a lattice matrix that consists of two projection matrices from the base lattice space to a tile-space and perp-space, respec- tively. In selecting points in the test window, the method of infinitesimal transfer of the test window is used and the boundary conditions of the test window are investigated. The QL obtained is composed of four kinds of pro- totiles, which are derived by choosing triplets out of seven basis vectors in the tile-space. The QL contains three kinds of dodecahedral clusters, which play an important role as packing units in the structure. A modification of the lattice matrix making the QL periodic in the z direction is also considered. © 2004 MAIK “Nauka/Inter- periodica”. 1 INTRODUCTION projection of vectors onto a plane perpendicular to the The discovery of icosahedral quasicrystal in 1984 fourfold axis forms an eightfold star with relative [1] stimulated both experimental and theoretical studies length 1/ 3 , which coincides with the configuration of in this field [2, 3]. The theoretical study of quasiperi- basis vectors of a 2D octagonal pattern, or Beenker’s odic tiling, which was developed before the discovery tiling [9]. The geometry of the seven basis vectors is of quasicrystals by de Bruijn [4] in connection with shown in Fig. 1. Penrose tiling, is based on the projection method. As methods of generating quasiperiodic tiling, the cut method and self-similar methods are known. The former is more general than the projection method and can generate a wider class of quasiperiodic tilings [5]. In this paper, a three-dimensional (3D) quasiperiodic tiling, or a quasi-lattice (QL), developed by the authors is presented and its method of generation and its prop- erties are explained—based on a 7 × 7 lattice matrix derived from a cuboctahedron and showing quasi- eightfold symmetry in one of its two-dimensional (2D) projections (cf. [6–8]). BASIS VECTORS, PROTOTILES, AND DODECAHEDRAL CLUSTERS A 3D QL based on a projection from a seven-dimen- sional (7D) hypercubic lattice (base lattice space) to a 3D space (tile-space) is proposed. The seven basis vec- tors in the tile-space are derived from the cuboctahe- dron and grouped into two sets of vectors of different lengths: three vectors along the fourfold axis of relative length 1/3 and four vectors along the threefold axis Fig. 1. Seven basis vectors derived from the normal of a of relative length 1/ 2 . These are chosen so that the cuboctahedron. They are grouped into two sets—three from the square facet and four from the triangular one—with a 1 This article was submitted by the authors in English. length ratio of 2/ 3 . 1063-7745/04/4904-0527 $26.00 © 2004 MAIK “Nauka/Interperiodica” 528 WATANABE et al. Table 1. Prototile parameters of a 3D quasi-lattice abccosα, (α, deg) cosβ, (β, deg) cosγ, (γ, deg) V C 1 1 1 0, (90) 0, (90) 0, (90°)1 6 3 3 2 S 1 1------- 0, (90) ------- , (54.7)------- , (54.7) ------- 2 3 3 2 6 6 1 3 3 P ------- ------- 1 --- , (70.5) ------- , (54.7)------- , (54.7) 1 2 2 3 3 3 6 6 6 1 1 1 R ------- ------- ------- –--- , (109.5) --- , (70.5)--- , (70.5) 2 2 2 2 3 3 3 The constituent prototiles of the QL are derived by (RPD), and a square parallelogram dodecahedron choosing three vectors out of seven basis vectors. The (SPD)—that are assembled by different prototiles can combinations of three basis vectors are restricted to be identified as the constituent clusters of the QL. These four cases that make independent prototiles. They are a dodecahedra consist of prototiles (RD = 4R, RPD = 2P + cube (C), a square parallelogram parallelepiped (S), a 2S, and SPD = C + 3S, respectively). The geometrical rhombic parallelogram parallelepiped (P), and a rhom- packing arrangements of these dodecahedra are illus- bohedron (R), as shown in Fig. 2 by a ball and stick trated in Fig. 3. The shapes of the facets of the constit- model. The cell constants of these prototiles are listed uent prototiles are identified by small letters: r for the in Table 1. rhombus, p for the parallelogram, and s for the square, Three kinds of dodecahedra—a rhombic dodecahe- with parentheses showing the facet as seen from inside. dron (RD), a rhombic parallelogram dodecahedron The number following the letter indicates the shape CP SR Fig. 2. Each figure includes a cuboctahedral star shown by balls and lines, with the prototile shown by dark balls. CRYSTALLOGRAPHY REPORTS Vol. 49 No. 4 2004 A QUASI-LATTICE BASED ON A CUBOCTAHEDRON 529 R (r2) (r1) (r2) R 2 r r2 r2 r1 (r1) r2 r1 r2 (r1) (r2) r2 r1 r2 (r2) r1 r2 r1 (r1) (r1) r2 (r2) r2 r2 (r2) r1 (r1) r2 R r2 R RD = 4R P (p2) S (r1) p1 p1 p2 s1 s1 p1 (p2) (s2) p1 (p1) r1 (p2) p1 p1 p1 (r1) p1 p1 (p2) r1 p2 s2 (s1) (p2) (p1) r1 p1 (p2) P S RPD = 2P + 2S S s1 (p2) (p2) p1 C (s2) s1 p1 (s1) s1 (s1) p2 p2 s1 s1 (p1) (s1) (s1) S s1 p2 p2 s2 s2 p2 s1 (p1) (p2) (p1) p2 p1 S (s2) SPD = C + 3S Fig. 3. Three kinds of dodecahedral clusters are shown (from top to bottom) in the right column and the expanded constituent pro- totiles in the left column. type, and matching is allowed only with the facet of the ing the proper transfer vector. Pleasants [12] proposed same shape type. The shapes of the facet of contact play a method—called the infinitesimal transfer of test win- an important role in defining the inflation rules in the dow to avoid overlapping or absence of prototiles—in self-similar method of QL construction. This problem which the test window is transferred infinitesimally in a is discussed in detail elsewhere [10, 11]. direction not parallel to any of the facets of the window and in which the projected points that stay inside both the original and the transferred window are selected. PROJECTION METHOD AND THE BOUNDARY OF THE TEST WINDOW Figure 4 shows the case in which a one-dimensional (1D) tiling is generated by the projection from a 2D lat- In the projection method of generating quasiperi- tice space. The axis x is the tile-space and x' is the perp- odic tiling, it is known that overlapping or absence of space. If the test window obtained by projecting a unit prototiles in the tiling is caused when a projected point square into the perp-space retains its initial position in perp-space falls on the boundary of the test window (zero transfer) and both ends are inside, the upper left and the boundary condition (open or closed) is not and lower right corners of the unit square are inside the properly specified. window and overlapping in the prototile occurs in the Since this problem is avoided by choosing an appro- tiling. By infinitesimal transfer of the test window priate transfer vector of the window, no discussion is along the axis of the perp-space, shown by an arrow (no usually made except in the case of the method of select- distinction being made in this case between the transfer CRYSTALLOGRAPHY REPORTS Vol. 49 No. 4 2004 530 WATANABE et al. x' the angle between vectors 1 and 4 and not eightfold symmetric with respect to the origin, although the shape of the test window is eightfold symmetric. QUASIPERIODIC LATTICE BASED ON A CUBOCTAHEDRON x The 7 × 7 lattice matrix A in a 7D space is given in (1) in which the upper three rows correspond to the pro- jection matrix to 3D tile-space (x, y, z) and the lower four rows correspond to that to 4D perp-space (x', y', z', w'): 1/ 2 1/ 2–1/2100 1/ 2 Fig. 4. Test window on perp-space x' for a 1D tiling. 1/ 2 1/ 2 1/ 2–010 1/ 2 1/ 2– 1/ 2 1/21/2001 1 and the infinitesimal transfer), the window becomes a A = -------–1/2 – 1/2 1/2–200 1/2 . (1) line segment that is closed at the upper end (shown by 3 a filled circle) and open at the lower end (shown by an –1/2 – 1/2 – 1/2 1/2 0 2 0 open circle) and the overlapping of prototiles is avoided. –1/2 1/2– 1/2 –002 1/2 Figure 5 shows the unit polygon (a), or the projec- 3/2–3/2 3/2 –3/2–000 tion of a four-dimensional (4D) unit cube to the tile- The seven column vectors by upper three rows of space, and the test window (b) of Beenker’s tiling with matrix A represent the basis vectors in the tile-space zero transfer.