Lattice

“ Ital. latta, Fr. latte) , a thin flat strip of wood or other material used in building to form a base or groundwork for plaster, or for tiles, slates or other covering for roofs. Such strips of wood are employed to form lattice-work, or for the bars of venetian blinds or shutters. A “lattice” (O. Fr. lattis) is an interlaced structure of laths fastened together so as to form a screen with diamond-shaped or square interstices. Such a screen was used, as it still is in the East, as a shutter for a window admitting air rather than light ”

“ The Sphæroid Skeletons or Lattice-Spheres.—The "lattice-spheres" or sphæroid skeletons are the simplest and most primitive forms of lattice shells, and are widely distributed in the three legions Spumellaria, Acantharia, and Phæodaria, whilst they are entirely wanting in the Nassellaria. The round lattice-shell is either a true sphere in the geometrical sense, or an endospherical polyhedron, i.e., a polyhedron, all whose angles lie in the surface of a sphere (§ 25) . ”

“ In the simplest example of the three-dimensional graph we have to solve the system
and a system for the general lattice constructed upon the same principle. The system has been discussed by MacMahon, Phil. Trans. vol. clxxxvii. A, 1896, pp. 619-673, with the conclusion that if the numbers of nodes along the axes of x, y, z be limited not to exceed the numbers m, n, l respectively, then writing for brevity 1 – xs = (s) , the generating function is given by the product of the factors
one factor appearing at each point of the lattice. ”

“ In the Nassellaria the regular lattice-structures generally exhibit hexagonal or circular meshes, whilst the irregular are either polygonal or roundish; the irregular forms are, however, much more abundant than the regular, and a further distinction from the Spumellaria consists in the fact that the primary skeletal elements, from which the lattice is secondarily developed, exercise a predominant influence upon their form. ”

“ According to the different positions to which the focus of the microscope is brought, the hexagonal tracery exhibits a different appearance; either a regular network of equal hexagonal frames, or a lattice with equal and equidistant circular openings, or a combination of three crossed and equidistant systems of parallel lines (crossed at angles of 120°) . ”

“ By using the valency volume instead of the molecular volume in the author’s formulæ for calculating the molecular distance ratios, which have been shown in Chapter X. to afford the relative dimensions of the unit cell of the space-lattice, and the distances of separation of the centres of gravity of the molecules from each other along the directions of the crystal axes, Pope and Barlow arrive at new ratios, which they term equivalence parameters. ”

“ For we pass here into the borderland between ordinary liquids—singly refractive and structureless, in which the molecules are rolling over each other with every possible orientation—and solid true crystals possessing homogeneous structure, and the basis of which is a space-lattice arrangement of the chemical molecules, determinative of the system of symmetry displayed. ”

“ It may be quite definitely stated, however, that there is a considerable amount of experimental evidence that the unit of the space-lattice of the crystal structure is certainly not more complex than the chemical molecule, the idea of an aggregation of chemical molecules to form a “physical molecule” acting as a structural unit having proved to be a misleading myth. ”

“ That which makes the Lawn so transparent, is by the Microscope, nay by the naked eye, if attentively viewed, plainly enough evidenced to be the multitude of square holes which are left between the threads, appearing to have much more hole in respect of the intercurrent parts then is for the most part left in a lattice-window, which it does a little resemble, onely the crossing parts are round and not flat. ”

“ Lattices are preferable to glazed lights, as they admit a free circulation of air; and if too much wind draws in, oiled paper may be pasted over the lattice, or a frame constructed so as to slide backwards and forwards at pleasure. ”

“ The pockets or lattice bars of the plates are for the purpose of supporting the active material which is of a weak and spongy nature. The active material on the positive plate is usually litharge, while that on the negative plate is red lead. ”

“ Icel. hlaða.
Latin, s. Latin, B 519.
Latis, s. lattice, T. ii. 615. (Many MSS. have gates; see note.)
Latitude, s. (1) breadth, A. i. 21. 27; (2) the breadth of a climate, or a line along which such breadth is measured, A. ii. 39. 19; (3) astronomical, the angular distance of any body from the ecliptic, measured along a great circle at right angles to the ecliptic, A. pr. 71; (4) terrestrial, the distance of a place N. or S. of the equator, A. ii. 39. 24; B 13, E 1797.
Latoun, s. latten, a compound metal, like pinchbeck, containing chiefly copper and zinc, A 699, 3251, C 351, F 1245 ”

“ Let’s examine the latticework of atoms that make up any solid material. In electrical insulators all the atoms’ outer electrons [5] are held tightly by valence bonds to the neighboring atoms. In contrast, any metal has many relatively loose electrons which can wander freely through its latticework. This is what makes metals good conductors. ”

“ Or we may quote the words of P. von Groth (British Assoc. Rep., 1904) : “A crystal—considered as indefinitely extended—consists of n 585 interpenetrating regular point-systems, each of which is formed of similar atoms; each of these point-systems is built up from a number of interpenetrating space-lattices, each of the latter being formed from similar atoms occupying parallel positions. All the space-lattices of the combined system are geometrically identical, or are characterized by the same elementary parallelopipedon.” ”

“ For it is not unknown that Lactantius, a writer of prominence in other lines although but little versed in mathematics, spoke very childishly about the form of the earth when he ridiculed those who declared that it was spherical. Hence it should not seem strange to the learned if some shall look upon us in the same way. Mathematics is written for mathematicians, to whom these labors of ours will seem, if I mistake not, to add something even to the republic of the Church.... ”

“ Here two crystals or individuals of the same kind have grown together in a certain symmetrical manner, such that one portion of the twin may be brought into the position of the other by reflection across a plane or by rotation about an axis. The plane of reflection is called the twin-plane, and is parallel to one of the faces, or to a possible face, of the crystal: the axis of rotation, called the twin-axis, is parallel to one of the edges or perpendicular to a face of the crystal. ”

“ If, for example, the spectrum of an atom, hitherto supposed to be elementary, give several colours, it is probable that there are several component atoms vibrating at different rates—corresponding to the colours—each atom having its own chemical characteristics. How simple the means by which results so momentous are achieved! Strange that an angular bit of glass, turning aside a ray of light, should be the means of revealing to us the secrets of the two extremes of the material universe—worlds and atoms! ”

“ In this way may we conceive that the rings are drawn out. The smaller bodies would be piecipitated most rapidly, and the transparency of the obscure would be accounted for, by supposing it composed of the finer particles. The precipitate is, however, constantly receiving new accessions, so that it is always becoming more conspicuous, and its recent detection may be due to its increasing density. The middle portion of the system we would expect to be the most dense; and we do find the inner bright ring to be the brightest. ”

“ The moment was critical for our interview. Could he do it? Were we the kind of person that he could give it to? Could we get it if he did?
“I think I can,” he said. “Let us begin with the assumption that the atom is an infinitesimal magnitude. Very good. Let us grant, then, that though it is imponderable and indivisible it must have a spacial content? You grant me this?”
“We do,” we said, “we do more than this, we give it to you.”
“Very well. If spacial, it must have dimension: if dimension—form. Let us assume ex hypothesi the form to be that of a spheroid and see where it leads us.” ”

“ The origin of cleavage in slate has been the subject of much speculation and investigation, but like many other problems it was solved through the invention and application of the microscope. Thin layers of slate have been made, the same as with limestone and chalk, so thin that the light would readily pass through it and that an examination of the particles could be readily made, showing their arrangement under varied conditions. Science is indebted to the microscope for the solution of very many problems that for ages before had puzzled philosophers. ”

“ The number of atoms transmitted by a layer, and remaining effective to produce weight in the next lower one, will bear the same ratio to the number reaching the first that the volume of the spaces or pores in the layer bears to its total volume. Assuming the body to be of uniform density, this ratio will be constant, and since the weight of each layer is proportional to the number of atoms available to collide with its substance, this ratio represents the relative weight of any layer to that next above It. ”

“ A purely geometrical question, of no importance in respect to the molecular tactics of a crystal but of considerable interest in pure mathematics, is forced on our attention by our having seen (§ 27) that a homogeneous assemblage of solids of given shape, each touched by twelve neighbours, has three freedoms which may be conveniently taken as the three angles specifying the orientation of each molecule relatively to rows of the assemblage as explained in § 30. ”

“ In short, if the meeting together and collecting of atoms would effect all things, all things would grow together in the air, since atoms flutter about through empty space. Why cannot the herb, why cannot the tree or grain, arise or be increased without earth, without roots, without moisture, without seed? From which it is evident that nothing is made up from atoms, since everything has its own peculiar and fixed nature, its own seed, its own law given from the beginning. ”

“ The sphere of change is universally found to be limited in such a way that any a changes only into α, α1, α2, . . ., any b only into β, β1, β2; and, in general, every real being changes only into such a ‘closed series’ of forms as, taken collectively, are deducible from the original nature of the being; while no being can ever pass out of the series of its own forms over into a series of forms belonging to another being. ”

“ Thus became slowly evolved the conception of a crystal as a group of similar planes symmetrically disposed around certain definite and obvious systems of axes, and so placed that the intercepts, or parameters, on these axes bore to each other a simple numerical ratio. Representing by a : b : c the ratio of the intercepts of a plane on the three axes of a crystal of a given substance, then the intercepts of every other plane of this, or of any other crystal of the same substance, conform to the general proportion m a : n b : p c, in which m, n, p are three simple whole numbers. ”

“ Mass, electric charge, temperature, have the same symmetry, of a type called scalar, that of the sphere. A current of water and a rectilineal electric current have the symmetry of an arrow, of the type polar vector. The symmetry of an upright circular cylinder is of the type tensor. All of the physics of crystals can be expressed in a form in which the particular phenomena in question are not specified, but in which are examined only the geometrical and analytical relations between the types of quantities where certain ones are considered as causes and the other as effects. ”

“ For that nothing is made by means of atoms, is apparent from this, that everything has a definite [14] seed, unless by chance we shall believe that the nature both of fire and water is derived from atoms. Why should I say, that if materials of the greatest hardness are struck together with a violent blow, fire is struck out? Are atoms concealed in the steel, or in the flint? ”

“ It is manifest that, unless the atoms had come together and made the heaven, the gods would still be suspended through the midst of empty space. By what counsel, then, by what plan, did the atoms from a confused mass collect themselves, so that from some the earth below was formed into a globe, and the heaven stretched out above, adorned with so great a variety of constellations that nothing can be conceived more embellished? ”

“ My subject this evening is not the physical properties of crystals, not even their dynamics; it is merely the geometry of the structure—the arrangement of the molecules in the constitution of a crystal. Every crystal is a homogeneous assemblage of small bodies or molecules. ”

“ Again, in some instances, we have more than three different planes of cleavage, as in Fluor Spar, where we have four. The solid, bounded by four planes, is a tetrahedron; or if we take four pairs of parallel faces, an octahedron. But if we attempt to take either of these forms for our integrant molecule, we are met by this difficulty: that a collection of such forms will not fill space. Perhaps this difficulty will be more readily conceived by the general reader if it be contemplated with reference to plane figures. ”