Kristallographie in Räumen beliebiger Dimensionszahl. I. Die Symmetrieoperationen

The symmetry operations occurring in lattices of n dimensions are listed. A symmetry operation is determined by the characteristic values S_[nu] of its secular equation. If all S_[nu] are algebraically conjugated, the symmetry operation is transitive. A transitive symmetry operation of multiplicity m can occur only in a space of n = [phi](m) dimensions, where [phi](m) is Euler's function denoting the number of co-prime residue classes of m. Intransitive symmetry operations have several classes of mutually conjugated S_[nu]. Each class represents a partial multiplicity m_k and transforms a partial space of n_k = [phi](m_k) dimensions in itself. The total multiplicity of an intransitive symmetry operation is the least common multiple of its partial multiplicities, its total number of dimensions n is the sum of all n_k. A symmetry operation is called a rotation or reflexion according to whether the partial multiplicity 2 occurs an even or odd number of times. The number of degrees of freedom is given by the number of partial multiplicities 1; it is equal to the number of dimensions of the subspace transformed in itself. The same number also determines the number of possible glide components, i.e. rational fractions of the translations in invariant directions with the multiplicity of the symmetry operation as denominator.