General three-dimensional point groups

Any solid object can be classified in terms of the collection of symmetry elements that can be attributed to the shape. The combinations of allowed symmetry elements form the general three-dimensional point groups or non-crystal-lographic three-dimensional point groups. The symmetry operators are described here by the International or Hermann-Mauguin symbols. 

A solid can belong to one of an infinite number of general three-dimensional point groups. However, if the rotation axes are restricted to those that are compatible with the translation properties of a lattice, a smaller number, the crystallographic point groups, are found. The operators allowed within the crystallographic... 

This illustrates a type of symmetry only seen in crystals and other extended arrays. That is, the symmetry operation combines both elements of point symmetry (as seen in molecules) and translation (which generates arrays). Here you can see that the repeat of this operation yields a vertical translation of one unit. The two-and three-dimensional space groups are realizations of the more general topic of group theory, which has been one of the tremendous scientific achievements in the last two centuries in the field of pure mathematics. 

Symmetry point groups specify the symmetry properties of a general three-dimensional object of finite extension and apply to all such objects. However, if the object consists of only a finite number of points, such as the vertices of a polyhedron or the atoms in a molecule, then further classification of... 

Now we come to a totally different method for producing matrix representations of a point group a method which involves the concept of a function space. The word space is used in this context in a mathematical sense and should not be confused with the more familiar three-dimensional physical space. A function space is a collection or family of mathematical functions which obeys certain rules. These rules are a generalization of those which apply to the family of position vectors in physical space and in order to help in understanding them, the corresponding vector rule will be put in square brackets after each function rule. 

Ogston s concept is a valid and important generalization. Three constraints are necessary to fix an object in three-dimensional space, but they need not all be points of attachment. In principle the two a groups of a molecule of the type Ca2bd could be distinguished by an enzyme if the three constraints were one point of attachment, one pocket, into which b could fit but d could not, and the position of the reactive groups of the catalytic site. 

This family of operators can be regarded as an extension of the family of point symmetry operators. Symmorphy is a particular extension of the point symmetry group concept of finite point sets, such as a collection of atomic nuclei, to the symmorphy group concept of a complete algebraic shape characterization of continua, such as the three-dimensional electron density cloud of a molecule. In fact, this extension can be generalized for fuzzy sets. 

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