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Crystallography, the science of crystalline form, is a branch of the geometry of solids applied to the elucidation of the forms actually exhibited by natural inorganic substances and the cognate bodies built up in the laboratory. The history of the development of this science having been sketched in the article Crystal (q.v.), we need here only give an outline of its principles as now understood. Though, as we then saw, optical, thermal, and electrical characters are intimately related to crystalline form, crystallography deals primarily with form. In form, as Steno practically showed, linear dimensions obey no law in natural crystals. Though their angles are determinate, they are not geometrically regular solids, their faces may be of any size. The primary characteristic of crystals, however, is symmetry; their particles are symmetrically arranged with reference to certain dominant lines or axes, so that their bounding-planes or faces make angles constant for each substance, and when, as is usual, they exhibit a tendency to splitting or cleavage, it is in directions symmetrically arranged as are their faces. The facts of crystallography are, therefore, mainly obtained by the measurement of the angles of crystals. The most mathematically satisfactory explanation of crystalline form is that elaborated by Professor W. H. Miller in 1839. In this system the position in space of any face of any crystal is determined by its relation to three axes of indefinite length, which intersect at a point called the origin within the crystal. As these axes are not all in one plane, every plane in the universe must cut at least one of the three, and its position can be represented by a symbol of three algebraical indices. All parallel planes on one side of the origin have the same symbol, which implies the principle that linear dimensions are subject to no law. One of Hauy's chief discoveries was that the intercepts or distances from the origin along the three axes at which faces cut the axes are in the ratios of whole numbers, generally less than seven. In each of "the six crystallographic systems some one plane is taken as a plane of reference, and is known as the parametral plane. It is one cutting all the three axes at intercepts, having a constant ratio in each system, and the simplest whole numbers expressing these ratios are called the parameters.

That the three crystallographic axes are not purely imaginary, but represent lines along which crystallisation has acted, is shown by the hollow, skeleton, or hopper-shaped crystals of rock-salt, and by such crystals as those pentagonal dodecahedra of pyrite from Traversella, in which each axis is terminated at each end by a similar but smaller dodecahedron.

The law of symmetry may even control two, four, or more distinct crystals aggregated together so that they mutually intersect symmetrically and form a twin-crystal, or macle, which will often have re-entering angles, or angles the apices of which point towards the crystal, as in the arrow-shaped macles of selenite, the cruciform ones of staurolite, and the three- or six-rayed forms of snow. These latter forms are reproduced in the "negative" crystals, or ice-flowers, hollows melted by Professor Tyndall from the centre of a block of ice by a beam of electricity.

On the other hand, there are partial exceptions to symmetry in what are termed hemihedral or mero-symmetrical forms, such as those cubes of boracite (q.v.), in which four out of their eight angles are truncated. Such forms exhibit differences of thermo-electric properties corresponding to these differences of form.

The six systems are - (1) The Qubic, tesseral or monometric, with the most perfect symmetry, nine planes of symmetry, its three axes at right angles and its three parameters equal, including the cube, as in rock-salt, fluor, and pyrite; the regular or equilateral octahedron, as in magnetite; the rhombic dodecahedron, as in garnet, and many other forms; (2) the Pyramidal, tetragonal or dimetric, including the right-square prism and square-based octahedron, with five symmetral planes, axes at right angles, and two parameters equal, exemplified by tinstone; (3) the Rhombohedral, or hexagonal, including the rhombohedron, with three planes of symmetry, the six-sided prism, with seven, the scalenohedron, and many other forms, with axes at equal but not right angles and equal parameters, and exemplified by calcite, quartz, beryl, etc.; (4) the Prismatic, rhombic or trimetric, with three symmetral planes, axes at right angles and unequal parameters, including rhombic prisms and octahedra with rhombic bases, as in sulphur; (5) the Oblique, or monoclinic, with one plane of symmetry, only one axis at right angles to each of the others, and unequal parameters, including oblique prisms, as in gypsum, augite, and hornblende; and (6) the Anorthic, or triclinic, with no plane of symmetry, axes all at unequal angles and parameters unequal, including certain doubly oblique forms, as in blue vitriol.