The law of symmetry is a law in the field of crystallography concerning crystal structure. The law states that all crystals of the same substance possess the same elements of symmetry. The law is also named the law of constancy of symmetry, Haüy's law or the third law of crystallography.
Definition The way in which the law of symmetry was originally defined by Haüy in 1815 was based on his law of decrements and his conception of crystals being assembled of tiny parallelepipeds (molécules intégrantes) stacked up in three dimensions without leaving any gaps. The modern definition of the law of symmetry is based on symmetry elements, and is more in the German dynamistic crystallographic tradition of Christian Samuel Weiss, Moritz Ludwig Frankenheim and Johann F. C. Hessel. Weiss and his followers studied the external symmetry of crystals rather than their internal structure. René Just Haüy first lectured about his law of symmetry in 1795 but it was not until 1815 that it was finally published. Haüy states the law as follows: "It consists in this, that any one method of decrement (décroissement) is repeated on all those parts of the nucleus of which the resemblance is such, that one can be substituted for the other by changing the position of this nucleus with respect to the eye, without it (the nucleus) ceasing to be presented in the same aspect" Later authors stated the law in clearer forms:
"The law of symmetry by René Just Haüy (1815): if the shape of a crystal is altered, corresponding parts (faces, edges, angles) of the crystal are simultaneously and similarly modified." "One of the most important results of Haüy's researches was the discovery of the law of symmetry, according to which when one form of crystallization is modified by its combination with other forms, all the similar parts, the edges, angles and faces, are always modified at the same time and in the same way. "The way in which nature produces crystals is always that of the greatest symmetry, in that opposite and corresponding parts are always equal in number, arrangement and shape." René Haüy, 1815. The law of symmetry stands out as one of the foremost contributions by Haüy. It is more an intuition than a true scientific law, but warned crystallographers on the importance of symmetry. This is one of its possible formulations: "A given type of decrement repeats itself on all the parts of the nucleus that are so similar that they can be substituted one for the other, when changing the position of this nucleus with respect to the eye. I call this [sic] parts identical."
Symmetry elements Haüy's method of building crystals from stacked parallelepipeds has been replaced in modern crystallography by three-dimensional lattices (Bravais lattices). The 32 crystallographic point groups combine the following symmetry elements.
Axis of symmetry
If a crystal has an axis of symmetry through its centre, such that the crystal can be rotated around the axis into a position where it appears identical to the starting position, then it has an axis of symmetry. A crystal may have zero, one, or multiple axes of symmetry but, by the crystallographic restriction theorem, the order of rotation may only be 2-fold, 3-fold, 4-fold, or 6-fold for each axis. An exception is made for quasicrystals which may have other orders of rotation, for example 5-fold. An axis of symmetry is also known as a proper rotation.
Plane of symmetry If a crystal can be divided by a plane into two mirror-image halves, then the plane is a plane of symmetry. A crystal may have zero, one, or multiple planes of symmetry. For example, a cube has nine planes of symmetry. A plane of symmetry is also known as reflection symmetry or mirror symmetry.
Centre of symmetry
If every face of a crystal has another identical face at an equal distance from a central point, then this point is called the centre of symmetry symbolised as i. A crystal can only have one centre of symmetry. A centre of symmetry is also known as point reflection, inversion symmetry, or centrosymmetry.
Rotoinversion symmetry
A rotoinversion, symbolised as (1, 2, 3, 4 or 6), is a combination of a rotation about an axis and a reflection in a plane perpendicular to that axis. As an example, a two-fold rotoinversion (2) is illustrated in the figure. Rotoinversion is also known as improper rotation, rotoreflection, or rotation-reflection.
History René Just Haüy showed in 1784 that the law of constancy of interfacial angles could be accounted for if a crystal were made up of minute building blocks (molécules intégrantes), such as cubes, parallelepipeds, or rhombohedra. Haüy's method is named the law of decrements. The law of rational indices was not stated in its modern form by Haüy, but it is directly implied by his law of decrements.
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