Tilings, coverings, clusters and quasicrystals

The projection method for constructing quasiperiodic tilings from a higher dimensional lattice provides a useful context for computing a quasicrystal's vertex configurations, frequencies, and empires (forced tiles). We review the projection method within the framework of the dual relationship between the Delaunay and Voronoi cell complexes of the lattice being projected. We describe a new method for calculating empires (forced tiles) which also borrows from the dualisation formalism and which generalizes to tilings generated projections of non-cubic lattices. These techniques were used to compute the vertex configurations, frequencies and empires of icosahedral quasicrystals obtained as a projections of the D 6 and Z 6 lattices to R 3 and we present our analyses. We discuss the implications of this new generalization.

Acta Crystallographica Section A Foundations of Crystallography, 1994

Aperiodic crystalline structures, commonly called quasicrystals, display a great variety of combinatorially possible local configurations. The local configurations of first order are the vertex configurations. This paper investigates, catalogues and classifies in detail the latter in the following important two-dimensional cases: the Penrose tiling, the decagonal triangle tiling and some twelvefold tilings, including the patterns of Stampfli, GS.hler, Niizeki and Socolar, as well as the square-triangle and the shield patterns. The main result is a comprehensive study of the three-dimensional primitive icosahedral tiling in its random version. All its 10 527 combinatorially possible noncongruent vertex configurations are constructed, coded, listed and classified. Methods for coding and representation of local configurations by formulae and diagrams, in particular those of Schlegel, are discussed. The paper also describes the algorithm used to generate them. The formal classification of local configurations by the characteristic integers rank, degree and order is also discussed. appearing in CBED patterns are given for all the (3 + 1)-dimensional space groups of the incommensurately modulated crystal.

Journal De Physique, 1990

2014 Nous présentons un nouvel algorithme pour la génération des structures quasicristallines. Il est relié à la méthode de coupe et projection, mais il permet une génération directement dans l'espace « physique » E de la structure. La sélection des sites dans l'espace orthogonal est remplacée par un test directement dans une grille de domaines d'acceptance dans l'espace E. Cette méthode montre qu'il y a une sorte de réseau cristallin sous-jacent au quasicristal. Nous illustrons la construction dans le cas 4D-2D avec les symétries d'ordre 5, 8, 10 et 12 qui sont obtenues par projection de 4D à 2D. Par la même méthode d'autres types de quasicristaux avec une symétrie plus basse, ayant un réseau moyen, sont construits. Nous présentons un exemple de symétrie 4. Les points de ce quasi-cristal sont un sous-ensemble des points du quasi-cristal ayant la symétrie complète d'ordre 8.

Physical review. B, Condensed matter, 1991

Current model networks for amorphous Ge contain five-membered rings and pentagonal dodecahedra to explain why in the radial distribution function the third peak of the diamond structure is missing. By presenting an algorithm based on a decoration of the three-dimensional Penrose quasilattice, we prove that this local pentagonal symmetry can be extended globally to an icosahedral quasicrystalline tetracoordinated network. Its structural elements and topological properties coincide with previous hand-built models of random networks. Thus it is suitable for simulating bulk properties of amorphous semiconductors.

Philosophical Magazine, 2006

Running head: Square-triangle inflation rules An inflation rule for square-triangle tilings is presented and its properties described. Its relationship to other previously found inflation rules, used to generate models of dodecagonal quasicrystals, is discussed. The rule is applied to generate a series of approximants of dodecagonal quasicrystal, including the well know n 3 Pm and P4 2 /mnm structures. The bigger the unit cell dimensions, the closer the structures of these approximants are to that of the dodecagonal quasicrystal.

2000

For each of the icosahedral patterns obtained by translating the strip used in the construction of the three-dimensional Penrose tiling we determine the isometries which leave it invariant. The variation of the pattern determined by a translation of the strip may describe some phase transitions or phasons in quasicrystals. Using our results one can determine the corresponding variation of the

Physical Review B, 1996

It is proposed that quasicrystal structure determination should include the calculation of cohesive energies using realistic potentials. A class of atomic decoration models for i-AlMn is then presented, adopting the ''canonical-cell'' tiling geometry, with ''Mackay icosahedron'' clusters placed on all its nodes. The remaining atomic positions are based, as far as possible, on the known structure of ␣-AlMnSi. These models guarantee good local packing of the atoms, whose displacements away from ''ideal'' positions are specified by only a moderate number of parameters. Certain atomic sites are uncertain as regards their occupancy and/or chemistry; variations of the decoration rules on these sites must be compared, in order to discover the correct one. Our models are well adapted to be relaxed under an effective Hamiltonian to optimize the cohesive energy; we show how the energies found in such relaxations can be used to extract an effective tile-tile Hamiltonian, as would be needed for future studies of phason elasticity and the development of long-range order. In addition, we clarify concepts needed for decoration models in general ͑in particular, the ways in which elaborate, more realistic decorations may be evolved from simpler ones͒. We also show that these decoration models are closely related, but not identical, to quasiperiodic structures defined using six-dimensional formalism.

Journal of Physics A: Mathematical and General, 1987

A quasiperiodic pattern (or quasicrystal) is constructed in real four-dimensional Euclidean space, having the noncrystallographic reflection group [3,3,5] of order 14400 as its point group. It is obtained as a projection of the eight-dimensional lattice E 8 , and has as a cross-section a three-dimensional quasicrystal with icosahedral symmetry.

Journal De Physique, 1989

2014 Au moyen de trois tuiles, nous construisons un pavage quasipériodique du plan, que nous relions au quasicristal octogonal. Ainsi, nous montrons que les coordonnées des n0153uds peuvent être obtenues de deux manières différentes. Le facteur de structure est calculé exactement. Ce pavage qui possède « presque » une symétrie d'ordre huit, soulève la difficulté de la détermination pratique de la symétrie d'un quasicristal. Finalement, nous montrons comment construire une large classe de pavage du type de l'octogonal, à partir de ce nouveau pavage.

Physical Review B, 1999

A likely mechanism for the formation of quasicrystals is by maximally covering space with overlapping, stable atomic clusters. This notion is here applied to the experimentally determined layered structure of octagonal MnSiAl quasicrystals, which can be described in terms of a decoration of the octagonal Ammann-Beenker tiling. This decoration is abstractly represented by a two-color version of the tiling. The covering cluster of the quasicrystal corresponds to an octagonal covering patch of the colored tiling. This covering patch appears in two variants with complementary colors. The three-dimensional quasicrystal has a centered octagonal translation module, and its space group is I8 4 /mcm. ͓S0163-1829͑99͒15525-5͔