Lattice of a crystal does indicate how to repeat the pattern of atoms. Base or motif indicates what to repeat. The sum of lattice and base form the crystal structure. Due to the systematic pattern of the atoms in a crystal, there are some typical relationships among the faces of crystals. The stereographic projections do aid in understanding the relationships among several crystal faces. Size and shape of the lattice of crystals determine the angular relationships among the crystal faces. Based on the regular arrangement of atoms, and the unit cells several crystallographic and mathematical relationships among the crystal faces were discovered.
The regular pattern of the lattices would lead to the uniform arrangement of faces and thus angles among them. The law of constancy of inter facial angles states that the angle between two faces remain constant measured at a certain temperature. Inter facial angles means, the angles between two faces. But in crystallography, the angles are measured between the normals drawn to any two adjacent faces of a crystal.
According to the law of axial ratios, the relative lengths of the crystallographic axes are measured with respect to the axis which is running from right to left. Except in the hexagonal system, the remaining systems, do possess three axes. Since all the three axes in the isometric system are of equal length, the axial ratio for them would be equal to one. Similarly, for the triclinic system, in which the axes are of variable length, the ratio too is variable.
Miller did measure the indices for various crystallographic forms. The procedure to measure the indices does involve measurement of parameters and reciprocating them, and clearing the fractions.
Twinning might occur during the growth of crystals. Twinned crystals resemble as if the one half is in the reverse direction of the other half. Twin plane is a plane which cuts the twinned crystal in such a way that one half is the mirror image of the other. Twin axis is an axis along which if the twinned crystal is rotated, it comes back to its untwinned state. Based on the twin plane, several types of twins were recognised. Albite twin, baveno twin, carlsbad, manebach, etc are various types of twinning and can be identified by their twin planes.
The demand for crystals in several fields lead to the invention of different types of methods of crystal growth such as the growth from liquid, from vapour, from solid, etc.