Three-dimensional pattern

Next, introduce two primitive concepts [bayes87a]. (1) An expansion, (C), of a Conway object C = (xi,yi,z = 0), which is a three-dimensional pattern, P, formed by copying all live cells of that object onto an adjacent plane, so that xi,yi,z = 0), xi,yy,z = 1). (2) A projection of a three-dimensional Life object into two dimensions, which exists if and only if (i) all of its live sites lie in two adjacent planes (say 2=0 and 2 = 1), and (ii) all pairs of sites, (xi,yi,0) and xi,yi, 1) are either both live or both dead. 

Crystals have definite geometric forms because the atoms or ions present are arranged in a definite, three-dimensional pattern. The nature of this pattern can be deduced by a technique known as x-ray diffraction. Ihe basic information that comes out of such studies has to do with the dimensions and geometric form of the unit cell, the smallest structural unit that, repeated over and over again in three dimensions, generates the crystal In all, there are 14 different kinds of unit cells. Our discussion will be limited to a few of the simpler unit cells found in metals and ionic solids. 

Greer J. Three dimensional pattern recognition an approach to automated interpretation of electron density maps of proteins. / Mol Biol 1974 82 279-301. 

The silicates are a large class of solids of great importance in industry as well as science, particularly geology. The prototype silicate is quartz consisting of Si04 tetrahedra which share their comers and edges and are arrayed in various three-dimensional patterns depending on the temperature. In other crystalline minerals the tetrahedra are linked in one-dimensional chains, or two-dimensional sheets. The arrays in these latter cases are combined with various metal ions. 

Petitjean, M. (1996) Three-dimensional pattern recognition from molecular distance minimization. J. Chem. Inf. Comput. Sci. 36, 1038-1049. 

Crystalline Solid solid in which atoms are arranged in definite three-dimensional pattern... 

The Band Theory of Crystalline Solids. Consider a crystalline solid. The atoms are arranged according to a three-dimensional pattern (or lattice) in which they have equilibrium interatomic distances. A thought experiment is now performed. The lattice is expanded, i.e., the interatomic distances are increased. 

In a plastic the polymer chains can be either intertwined simply at random to form a three-dimensional pattern, or they can be linked together by chemical bonds. This latter type is called a crosslinked polymer, and it... 

Periodic repclitions of a space lattice cell in three dimensions from the original cell vvill completely partition space without overlapping or omissions. El is possible to develop a limited number of such three-dimensional patterns. Bravais. in 1848. demonsirated geometrically that there were but fourteen types of space lattice cells possible, and that these fourteen types could be subdivided into six groups called systems. Each system may be distinguished hy symmetry features, which can be related lo four symmetry elements ... 

By linking the centres of the spheres in a three-dimensional packing you can create a three-dimensional pattern of centres, a so-called space lattice. Such a lattice consists of many elementary cells. An elementary cell is the smallest possible spatial unit in the crystal lattice which is repeated in three directions (according to the mathematical x-, y-and z-axes) in the lattice. Figure 4.6 shows a space lattice and an elementary cell. 

Another X-ray structure reported is that of ( )-colchicine which shows unusual physical properties a melting point of 280-282°C and a much lower solubility than that of the natural colchicine (1) in commonly used solvents (5). The X-ray data showed an extensive three-dimensional pattern of hydrogen bonding, linking chains of alternate D and l molecules related by a glide plan and hydrogen bonded by the interaction... 

Dinca and colleagues (2008) have combined fibril self-assembly, lithographic features, and covalent and noncovalent binding to create three-dimensional patterned fibrillar structures that could be used in electronics or tissue engineering. Images of these structures, which are described as fibril bridges, are shown in Figure 15. 

The studies outlined in this section show a number of different techniques that can be used to align fibrils and other peptide nanofibers in order to create micron-sized patterns from these self-assembling materials. This section also explored some of the covalent and noncovalent strategies for attaching fibrils to surfaces, and how these methods may be exploited to create three-dimensional patterns of fibrils that have an exciting range of potential applications. 

The crucial point is the idea that a body plan is simultaneously a phenotypic structure and a deposit of information. If information could be transported without three-dimensional structures, there would be no need to conserve three-dimensional patterns, but the information of a body plan is precisely about spatial organisation, and cannot be preserved without the three-dimensional structures which define that organisation. Traditional theories, in conclusion, have regarded the body plan exclusively as a phenotypic structure, not as a deposit of information (a supracellular memory), and it is this which has prevented them from explaining the conservation of the phylotypic stage. 

Such symmetry elements are not possible in crystals of cydodextrins, proteins, and nucleic acids, which all contain asymmetric carbon atoms, and therefore cannot be organized into highly symmetrical three-dimensional patterns. Natural product molecules of this sort crystallize in space groups in which the cyclic hydrogen-bonding motifs are formed without symmetry limitations. In this respect they resemble the lower symmetry amine hydrates described in Part IV, Chapter 21. 

A crystal may be explicitly defined as a homogeneous solid consisting of a periodically repeating three-dimensional pattern of particles. Mathematically, there are three key structural feamres to crystals. 

Wu and Woo [26] compared the isothermal kinetics of sPS/aPS or sPS/PPE melt crystallized blends (T x = 320°C, tmax = 5 min, Tcj = 238-252°C) with those of neat sPS. Crystallization enthalpies, measured by DSC and fitted to the Avrami equation, provided the kinetic rate constant k and the exponent n. The n value found in pure sPS (2.8) points to a homogeneous nucleation and a three-dimensional pattern of the spherulite growth. In sPS/aPS (75 25 wt%) n is similar (2.7), but it decreases with increase in sPS content, whereas in sPS/PPE n is much lower (2.2) and independent of composition. As the shape of spherul-ites does not change with composition, the decrease in n suggests that the addition of aPS or PPE to sPS makes the nucleation mechanism of the latter more heterogeneous. 

Pharmacophore Models. It is often useful to assume that the receptor site is rigid and that structurally different drugs bind in conformations that present a similar steric and electronic pattern, the pharmacophore. Most drugs, because of inherent conformational freedom, are capable of presenting a multitude of three-dimensional patterns to a receptor. The pharmacophoric assumption led to a problem statement that logically is composed of two processes. F irst is the determina-... 

Figure 3.44. OMAPs generated for two molecules can be logically intersected to determine which three-dimensional patterns are common.

It has been found by experiment that every crystal consists of atoms arranged in a three-dimensional pattern which repeats itself regularly. In a crystal of copper all of the atoms are alike, and they are arranged in a way called cubic closest packing, shown in Figures 3-3 and 3-4. This is a way in which spheres of uniform size may be packed together to occupy the smallest volume. 

Daniel C. G. and Spear F. S. (1998) Three-dimensional patterns of garnet nucleation and growth. Geology 26, 503 -506. 

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