Tiling Patterns Research Articles

QUASI2D: a program written to demonstrate quasiperiodicity and phason fluctuation

A program is described that demonstrates in two dimensions the concepts of quasiperiodic tiling and phason fluctuations. Tiling patterns having perfect fivefold or eightfold quasiperiodicity are displayed on the computer screen at any one of three chosen scales. These patterns are then progressively altered in real time by the application of phason flips to obtain disordered tiling patterns. Energy parameters may be specified to allow preference for different tile-pair combinations in the resulting distributions. In this way, with different combinations of the energies, distributions varying from pure random tiling to ones in which more ordered microdomains are to be seen may be obtained in real time via Monte Carlo simulation.

IDEAS

The “IDEAS” section for this month focuses on patterning using geometric figures. The sequentially planned activities help students discover the attributes—segments and angles—of various figures and how the figures can be combined to make a pattern; they then apply this information to the everyday world in such activities as making quilts and tessellations (tiling patterns). With the teacher's guidance, students develop the language needed to talk about their discoveries (definitions of less familiar terms are furnished in the Glossary): thus they will be able to communicate mathematically about geometric figures and patterning. The parent-child activity reinforces the school activity. Parents are invited to help students look for patterns in their community and share their findings with their child's class.

The inflation pattern of the three-dimensional Penrose tiling

The variety of different decorations of the inflation tiles of the three-dimensional Penrose pattern is derived. This is mainly caused by differing orientations of the rhombic icosahedra, which are placed between two stars at the vertices of the large rhombohedra. The derivation was enabled by the investigation of the possible positions of Ammann planes.

Dürer, Gaskets and Barnsley's Chaos Game

AbstractMichael Barnsley has generated the Sierpinski triangle or gasket using his “Chaos Game”, a technique for generating random points within a fractal object. Extensions of the method are discussed to generate fractal “gaskets” based on other regular polygons. One such figure resembles a tiling pattern based on pentagons known to Albrecht Dürer almost 500 years ago.

The magical Fibonacci number

Fibonacci numbers are explained, and some of the many manifestations of the Fibonacci series in nature are described. These range from the so-called golden spiral to the Penrose tiling patterns that describe the structure of quasicrystals. >

Deterministic chaos, fractals, and quantumlike mechanics in atmospheric flows

The complex spatiotemporal patterns of atmospheric flows that result from the cooperative existence of fluctuations ranging in size from millimetres to thousands of kilometres are found to exhibit long-range spatial and temporal correlations. These correlations are manifested as the self-similar fractal geometry of the global cloud cover pattern and the inverse power-law form for the atmospheric eddy energy spectrum. Such long-range spatiotemporal correlations are ubiquitous in extended natural dynamical systems and are signatures of deterministic chaos or self-organized criticality. In this paper, a cell dynamical system model for atmospheric flows is developed by consideration of microscopic domain eddy dynamical processes. This nondeterministic model enables formulation of a simple closed set of governing equations for the prediction and description of observed atmospheric flow structure characteristics as follows. The strange-attractor design of the field of deterministic chaos in atmospheric flows consists of a nested continuum of logarithmic spiral circulations that trace out the quasi-periodic Penrose tiling pattern, identified as the quasi-crystalline structure in condensed matter physics. The atmospheric eddy energy structure follows laws similar to quantum mechanical laws. The apparent waveparticle duality that characterizes quantum mechanical laws is attributed to the bimodal phenomenological form of energy display in the bidirectional energy flow that is intrinsic to eddy circulations, e.g., formation of clouds in updrafts and dissipation of clouds in downdrafts that result in the observed discrete cellular geometry of cloud structure.

Diffraction from quasi-crystals and disordered twinned aggregates

Following the discovery of systems with diffraction patterns exhibiting icosahedral symmetry by Shechtman, Blecht, Gratias & Cahn [Phys. Rev. Lett. (1984), 53, 1951–1953], which is forbidden in classical crystallography, interest has centred on Penrose tiling and its generalizations as a model for the explanation of these patterns, although Pauling [Nature (London) (1985), 317, 512–514; Phys. Rev. Lett. (1987), 58, 365–368] has proposed an alternative explanation in terms of crystal twinning. In an attempt to reconcile these two points of view a computer algorithm has been developed, following the methods outlined by Wolny, Pytlik & Lebech [J. Phys. C (1988), 21, 2267–2277], to grow a number of different structures which all tile the two-dimensional (2D) plane without defects. With the same two types of basic unit, it is shown that it is possible, on the one hand, to produce structures similar to the Penrose tiling pattern and, on the other, different types of twinned and disordered structures, with a continuous sequence of intermediates. Optical diffraction patterns of these structures have been obtained for comparison with real quasi-crystals.

The Design Of Choropleth Shadings For Maps On 2-And 4-Bit Color Graphics Monitors

Guidelines for the design of shading sequences for quantitative choropleth maps are fairly well established, having been the subject of a number of empirical studies. The most effective designs comprise monochrome shadings which vary in value from light to dark and part-spectral combinations which also incorporate a gradient of values. Neither of these alternatives are available on most moderately-priced microcomputer systems due to the very limited range of hues and brightnesses available on 2- and 4-bit CRTS. A study is reported here which evaluated three sets of choropleth shadings that would be available with such monitors. The sets consisted of two or three solid hues and tile patterns made up of the same hues. Subjects were asked to identify regions on choropleth maps where the mapped phenomenon was concentrated, and the accuracy and time required for them to answer was analyzed. Two of the shading methods proved to be acceptable, but the third was not.

High‐resolution electron microscopy of Al‐Mn‐Si icosahedral and Al‐Mn decagonal quasicrystals

SUMMARYAl‐Mn‐Si icosahedral and Al‐Mn decagonal quasicrystals have been studied by high‐resolution electron microscopy. High‐resolution micrographs of the Al‐Mn‐Si icosahedral quasicrystal taken with the incident beam along the five‐, three‐ and two‐fold symmetry axes are well interpreted on a three‐dimensional Penrose tiling model. An image of the Al‐Mn decagonal quasicrystal with the incident beam parallel to the ten‐fold symmetry axis shows a pattern of an aperiodic tiling with two types of rhombs which can be distinguished from the Penrose tiling using matching and inflation rules. The Al‐Mn decagonal quasicrystal examined coexists coherently with that of icosahedral symmetry exhibiting a definite crystallographic orientation relationship.

高分解能電子顕微鏡でみた準結晶の構造

The structures of icosahedral and decagonal quasicrystals are discussed from highresolution electron microscope observations of Al-Mn-Si icosahedral and Al-Mn decagonal quasicrystals. A high-resolution micrograph of the Al-Mn decagonal quasicrystal taken with the incidence parallel to the 10-fold symmetry axis shows a pattern of an aperiodic tiling with two types of rhombs, which is distinguished from the Penrose tiling with matching and inflation rules. An image of the Al-Mn-Si icosahedral quasicrystal is interpreted by the projection of the 3-dimensional Penrose tiling. In the Al-Mn-Si icosahedral quasicrystal, quench-ed phason strains and dislocations were obserbed.

Quasiperiodic tiling in two and three dimensions

Algebraic criteria which allow the vertices of two- and three-dimensional Penrose tiling patterns to be specified are presented. The application of these criteria to the interpretation of electron microscope observations from quasicrystalline structures is discussed.

Symmetry and Tessellations from Rotational Transformations on Transparencies

The title of this article may be a bit ostentatious, but it describes a simple yet rather dramatic procedure for producing symmetrical tessellated patterns. I feel that mathematics is all around us if we would only care to look. It is easy to remind students of mosaic tiling patterns, hexagonal or other interlocking paving patterns, or Escher's work. Recently I discovered that simple dot grids in a square (fig. I) or in a triangle (fig. 2) can produce tessellations (interlocking patterns that cover a surface).

Cultural continuity and the growth of West Indian religion in Britain

The change in patterns of religious adherence and ritual activity among migrants from peasant to urban industrial societies has been frequently noted by social scientists. However, tile pattern of religious adherence among West Indian migrants to Britain poses certain anomalies for the standard view of the nature and causes of such changes, going far beyond the commonly cited exception of the growth of Ras Tafari among black youth in Britain. At first sight, the change in religious adherence among West Indians following migration to Britain would seem to conform to the pattern familiar to social scientists--that migration from a peasant to an industrial society, whether within or across national boundaries, is associated with a decline in religious observance. This decline, which occurs particularly among men (Mol, 1972), may be reversed on return to the culture of origin (Christian 1972). The principal causes of this relationship between migration and religious decline have been held to be the relatively more open social structure of industrial societies--which encourages mobil i ty--and the anonymity .of urban life. Together, they tend to inhibit the development of moral communities, through which conformity to the practice of collective rituals is achieved (Hill, 1971a). Conversely, the conformity of the migrant to patterns of religious observance prevailing in the secularised host society has been seen as an aspect ofassimilation to dominant behaviour, and as a product of the migrant's perception of the host society as an amoral, individualised world in which religiously sanctioned norms of proper behaviour are suspended for the period of migration (Christian, 1972). Estimates of the extent of decline in religious observance among West Indians consequent upon migration to Britain vary greatly, probably because they are based on small-scale studies in different localities, but there seems to be no doubt that decline has occurred (Pearson, 1978, Hill, 1971a).

Let's Do It: Concepts, Art, and Fun from Simple Tiling Patterns

A tiling pattern (or tessellation) is a repetitive pattern, something like a mosaic, using one or more geometric shapes. The tiles in a tiling pattern should fit together so there are no holes or gaps between them. They are placed in a repeated pattern so that once begun, a huge plane surface could be covered by simply repeating the pattern over and over.

An evaluation of factors associated with the depression of immunity in malnutrition and in measles

In a previous communication (I) attention was drawn to the effects of protein-calorie malnutrition (PCM) on immunological responses in children and the way in which this may alter tile pattern and response to infection. In this further study is reported: 1) the pattern of immunological ehabiitation during recovery from PCM and how this correlates with tile biochemical pattern of recovery; 2) the role of some specific nutritional deficiencies in PCM in depressing immunity; 3) the proportion of bursal (“B”) lymphocytes in the peripheral blood and depression of humoral immunity; and 4) the role of infection (as represented by measles) in depressing immunity and how this compares with the depression due to PCM.

Bar Coliseo, en Sevilla

This bar is situated inside the «Coliseo» building, which houses a cinema, as well as a number of commercial establishments. In order not to break the unity of the total project, no attempt has been made to alter the exterior aspect of the bar. No attempt was made, either, to make it into an intimate, club type of bar, now so much in fashion. Rather has it been given a diaphanous style, seeking the best possible use of the floor space. The windows of the building are elongated, and there is an intermediate floor level, whose detailed structure is metallic. A cleverly designed staircase, of folded sheet metal connects the ground floor, the intermediate floor level and the restaurant. Materials were carefully chosen in accordance with their function. The colour scheme has a sustained unity throughout the building, and care has been taken to avoid surprising or vivid chromatic patterns. Ceramic enamels by the painter Santiago del Campo provide a feature of decoration on the ground floor, and also serve to cover up the return air ducts. On the top floor, the restaurant is fitted with coloured tile facings, the work of the Seville painters Maria Josefa Sanchez, Maria Dolores Sanchez and Emilio Garcia Ortiz. The bottom joints of the timber beams, in conjunction with the tile patterns, is reminiscent of the traditional Sevillian habit of placing ceramic units between the timber framework of buildings. The initial problem of the architect was to combine the optimum functional efficiency and aesthetic quality of the project, and the final solution is undoubtedly successful.