His earlier papers were mostly concerned with **crystallography**, and the reputation they gained him led to his appointment as Privatdozent at Konigsberg, where in 1828 he became extraordinary, and in 1829 ordinary, professor of mineralogy and physics.

The development of the theory of crystal structure, and the fundamental principles on which is based the classification of crystal forms, are treated in the article **Crystallography**; in the same place will be found an account of the doctrine of isomorphism, polymorphism and morphotropy.

In the article **Crystallography** the nature and behaviour of twinned crystals receives full treatment; here it is sufficient to say that when the planes and axes of twinning are planes and axes of symmetry, a twin would exhibit higher symmetry (but remain in the same crystal system) than the primary crystal; and, also, if a crystal approximates in its axial constants to 'a higher system, mimetic twinning would increase the approximation, and the crystal would be pseudo-symmetric.

In **crystallography** the icosahedron is a possible form, but it has not been observed; it is closely simulated by a combination of the octahedron and pentagonal dodecahedron, which has twenty triangular faces, but only eight are equilateral, the remaining twelve being isosceles (see **Crystallography**).

A point which divides a line, or a line which divides an angle, into two equal parts; in **crystallography** it denotes the bisector of the angle between the optic axes.

The mensuration of the cube, and its relations to other geometrical solids are treated in the article Polyhedron; in the same article are treated the Archimedean solids, the truncated and snubcube; reference should be made to the article **Crystallography** for its significance as a crystal form.

Bevelment, as a term of **crystallography**, means the replacement of an edge of a crystal by two planes equally inclined to the adjacent planes.

But they may, in accordance with the principles of **crystallography**, also occur in other forms symmetrically derived from the octahedron, - for example, the cube, the 2-faced figure known as the rhombic dodecahedron (fig.

In **crystallography** as " twins."

In **crystallography**, the regular or ordinary dodecahedron is an impossible form since the faces cut the axes in irrational ratios; the "pentagonal dodecahedron" of crystallographers has irregular pentagons for faces, while the geometrical solid, on the other hand, has regular ones.