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Five-Fold Symmetry in Crystalline Quasicrystal Lattices Proc. Natl. Acad. Sci. USA Vol. 93, pp. 14271–14278, December 1996 Colloquium Paper This paper was presented at a colloquium entitled ‘‘Symmetries Throughout the Sciences,’’ organized by Ernest M. Henley, held May 11–12, 1996, at the National Academy of Sciences in Irvine, CA. Five-fold symmetry in crystalline quasicrystal lattices DONALD L. D. CASPAR† AND ERIC FONTANO‡ Institute of Molecular Biophysics, Florida State University, Tallahassee, FL 32306-3015 ABSTRACT To demonstrate that crystallographic meth- example, such procedures have been applied by Steurer and his ods can be applied to index and interpret diffraction patterns colleagues to calculate five-dimensional Fourier maps from from well-ordered quasicrystals that display non-crystallo- three-dimensional x-ray diffraction patterns of decagonal- graphic 5-fold symmetry, we have characterized the properties phase aluminum-transition metal alloy quasicrystals (7–9). of a series of periodic two-dimensional lattices built from Projections from these physically abstract five-dimensional pentagons, called Fibonacci pentilings, which resemble ape- constructs produce real space maps, which show correlations riodic Penrose tilings. The computed diffraction patterns with the crystallographically determined atomic arrangements from periodic pentilings with moderate size unit cells show in related periodically ordered alloys (10–12). The success of decagonal symmetry and are virtually indistinguishable from this five-dimensional quasicrystallographic analysis suggests that of the infinite aperiodic pentiling. We identify the vertices that, because the diffraction data is only observable in three- and centers of the pentagons forming the pentiling with the dimensional reciprocal space, more conventional crystallo- positions of transition metal atoms projected on the plane graphic analysis might be applied to refine real space models perpendicular to the decagonal axis of quasicrystals whose of the atomic arrangements in these quasicrystals. structure is related to crystalline h phase alloys. The char- Quasicrystals are, by definition, aperiodic lattices. The acteristic length scale of the pentiling lattices, evident from diffraction pattern from one portion of such a lattice is the Patterson (autocorrelation) function, is ;t2 times the indistinguishable from that of another portion. A representa- pentagon edge length, where t is the golden ratio. Within this tive portion of a quasicrystal lattice can be chosen as a large distance there are a finite number of local atomic motifs whose unit cell of a perfectly periodic lattice, which would yield the structure can be crystallographically refined against the same diffraction pattern as the aperiodic lattice. A great experimentally measured diffraction data. variety of such periodic lattices can be constructed by selecting different portions of the aperiodic lattice as the unit cell. The Five-fold symmetry has been associated with magic and mys- fact that such lattices exist suggests that one member of this ticism since ancient times. Kepler, in his Mysterium Cosmi- class might be transformed into any other member by localized graphicum, published 400 years ago, described how he inge- displacive rearrangements of the constituent atoms. niously found the symmetry of the five Platonic polyhedra in Our surmise is that quasicrystals with icosahedral or decag- the structure of the solar system. Book II of his Harmonices onal symmetry may be modeled by periodic packing arrange- Mundi (1), on the congruence of harmonic figures, is a ments of icosahedra or pentagons in moderate-size unit cells pinnacle in the history of geometry, combining imaginative that can be locally rearranged, conserving key bonding rela- mathematical mysticism with profound insights into the sym- tions, to generate aperiodic lattices. In this paper, we focus on metry of polyhedra and polygonal tilings of the plane. Kepler’s regular arrangements of pentagons in the plane, applying the exploration of orderly arrangements of plane pentagons has same sort of packing rules as used by Du¨rer (13), Kepler (1), been viewed (2) as an anticipation of Penrose’s aperiodic and Penrose (3) in their explorations of pentagonal tilings. The tilings (3), which have served as models for the geometry of designs of these regular pentagonal tilings are related to the quasicrystal structures. arrangement of transition metal atoms projected on the plane Quasicrystallography has developed into an elaborate dis- perpendicular to the axes of local 5-fold symmetry in the alloys cipline since 1984 when Shechtman et al. (4) first reported with aluminum of the crystallographically regular h phase crystal-like diffraction patterns with forbidden icosahedral (10–12) and the decagonal quasicrystals (7, 8). symmetry from aluminum–manganese alloys, and Levine and Steinhardt (5) coined the name quasicrystals for the class of Graphics Methods quasiperiodic structures. Exposition of the results of many experimental studies on these novel alloys, and of the efforts To visualize the regular arrangements of pentagons (pentilings), of physicists to model their properties are presented in the their relation to crystal structures, quasicrystal diffraction pat- book Quasicrystals: A Primer, by Janot (6); the mathematical terns and Patterson functions, special purpose graphics routines concepts involved in the construction of aperiodic lattices are were developed. All images were created and rendered using described in Quasicrystals and Geometry, by Senechal (2). unique code in the POSTSCRIPT language (14). Pentilings were In their endeavors, quasicrystallographers have used a va- created using recursive routines, and coordinates needed for riety of mathematically sophisticated but physically unrealistic Fourier analyses were generated from the POSTSCRIPT code using models to analyze aperiodic lattices with icosahedral or de- the Aladdin Ghostscript interpreter. Once in Protein Data Bank cagonal symmetry. Quasicrystal structures have been repre- (PDB) format, the coordinates were used with the CCP4 package sented as projections into two- or three-dimensional space (15) to calculate electron density maps, structure factors, and from periodic models in five- or six-dimensional space. For Patterson maps. The maps were converted to grayscale images and then embedded in POSTSCRIPT documents. The construction The publication costs of this article were defrayed in part by page charge payment. This article must therefore be hereby marked ‘‘advertisement’’ in †e-mail: [email protected]. accordance with 18 U.S.C. §1734 solely to indicate this fact. ‡e-mail: [email protected]. 14271 Downloaded by guest on September 25, 2021 14272 Colloquium Paper: Caspar and Fontano Proc. Natl. Acad. Sci. USA 93 (1996) of twinned lattice images and montages used the POSTSCRIPT Du¨rer’s P0 5 2 pentiling appears in various guises in the clipping and superposition capabilities. structure of matter. For example, this pattern was found by Kiselev and Klug (16) in the cylindrical surface lattices formed Pentilings by pentamers of papovavirus coat proteins, which they called pentamer tubes. In the 72 pentamer icosahedral virus capsids We define a pentiling as an arrangement of regular pentagons of this oncogenic family (17), the 12 pentavalent pentamers in the plane in which each pentagon makes edge-to-edge make edge-to-edge contacts with their neighbors, as in the contact with two, three, four, or five neighbors, thereby sharing pentagonal dodecahedron; but the 60 hexavalent pentamers vertices in such a way that no gaps large enough to contain use, in addition to an edge-to-edge contact, one overlapped another pentagon are left in the array. A periodic pentiling is corner and two point-to-point contacts. In various polymor- a regular lattice with P pentagons in the unit cell. P is called phic aggregates of polyoma virus pentamers (18), different the pentile number. combinations of these adaptable contacts occur. Thus, the The simplest and most compact periodic pentiling is the first polymorphic packing of these virus pentamers is too complex tile pattern formed by pentagons described by Du¨rer (13) in A to be analyzed using the simple pentiling notions that are Manual of Measurements of Lines, Areas, and Solids by Means appropriate for decagonal quasicrystals. of Compass and Ruler, which he published in 1525. This tile pattern (which was illustrated by Du¨rer in his figure 24) is Crystalline h Phase Alloys shown in Fig. 1a. The pentile number for this lattice is P0 5 2, the zero subscript indicating that this is the fundamental The arrangement of transition metal atoms in the crystalline, member of its class. h phase alloys with aluminum can be modeled in two- The repeating motif of the P 5 2 pentiling (Fig. 1a) consists 0 dimensional projection by Du¨rer’s P 5 2 pentiling. Fig. 2 of the two regular pentagons of edge length E and the 368 0 illustrates the projected atomic structure of the FeAl inter- lozenge gap, also with edge length E. The crystallographically 3 metallic compound determined by Black (10). All the crystals defined unit cell is the parallelogram with short axis a0 5 tE, 2 he examined were twinned. One of the twinning arrangements long axis b0 5 t E, and included angle g 5 1088, where t is the 2 21 1 he inferred from his atomic model (11) is illustrated at the right golden mean (t 5 t 2 1 5 t 1 1 5 ⁄2(=511) 5 2 Cos 368 5 (2 Sin 188)21 5 1.618034. ). The unit cell can also be side of Fig. 2. The regular pentiling does not perfectly fit the represented by the 368 lozenge of edge length t2E. Because the map of Black’s projected unit cell because the ratio of the long ratio of the edge length of the unit cell lozenge to that of the to short axes from his measurements is 1.6108 6 0.0005 rather gap is t2, the fraction of the unit cell area occupied by than the expected golden ratio 1.6180, and his included angle 24 pentagons, defined as the packing density, is r0 5 (1 2 t ) 5 is 179 short of the expected 1088. These discrepancies are so 0.854102.
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