About the concept and definition of “noncrystallographic symmetry”

Abstract

The definition of "noncrystallographic symmetry" given in Volume B of the International Tables for Crystallography actually corresponds to the concept of "local symmetry". A new definition of "noncrystallographic symmetry" is proposed, which fully complies with that of "crystallographic symmetry" in Volume A of the International Tables for Crystallography.The concept of "noncrystallographic symmetry" is, quite obviously, directly related to that of "crystallographic symmetry" so that once a definition of the latter is given, that of the former is obtained spontaneously. This is probably the reason why no explicit definition of "noncrystallographic symmetry" is given in Volume A of the International Tables for Crystallography. Even the concept of noncrystallographic point groups, presented in section 10.1.4, is introduced without an explicit definition but only as the groups differing from the crystallographic point groups presented in Sections 10.1.2 and 10.1.3.As a matter of fact, one should not even need an explicit definition of "noncrystallographic symmetry" once that of "crystallographic symmetry" is given. Unfortunately, in the literature, and especially in the structural biology literature, the term "noncrystallographic symmetry" is used in a manner in striking contrast with what is directly implied by the definition of "crystallographic symmetry". This contradiction is so fundamental that it causes serious misunderstandings and misinterpretations. We will show that the use of "noncrystallographic symmetry" in structural biology is inconsistent with the accepted definition of "crystallographic symmetry" and suggest that an alternative terminology should be used.To understand the problem let us start with the definition of "crystallographic symmetry operations" given in [1] Section 8.1.5:A motion is called a crystallographic symmetry operation if a crystal pattern exists for which it is a symmetry operation.A motion is an isometry, a transformation keeping angles and distances unchanged, i.e. a transformation without deformation. A crystal pattern is the extension of a crystal structure to a periodic arrangement of whatever object, concrete or abstract, constitutes the structure. The atoms forming a crystal structure represent a special case of a crystal pattern. This definition applies to the n-dimensional Euclidean space E n and can be expressed in a quantitative way with the aid of group theory. With respect to a basis of E n , the symmetry operations of the space group of the crystal pattern are represented by (n þ 1)  (n þ 1) augmented matrices where the top-left n  n block represents the linear part of the operation (the part that leaves the origin fixed) and the additional column represents the vector part of the operation (the part which gives the translation component of the symmetry operation). The above definition of a crystallographic symmetry operation implies that with respect to a suitable basis of E n (namely a primitive basis of the periodic pattern) the linear part of the matrix represen...

From patterns to space groups and the eigensymmetry of crystallographic orbits: a reinterpretation of some symmetry diagrams in IUCr Teaching Pamphlet No. 14

“…The space group of the structure G(S) is, instead, the intersection of the eigensymmetries of these orbits. In fact, for each orbit, only the symmetry operations that are common to the other orbits are promoted to symmetry operations of the whole structure, the others being local symmetry operations [for the meaning of a local operation, see Nespolo et al (2008)…”

Symmetry and chirality in crystals

“…It was assumed that the two enantiomers must occur in the asymmetric unit and be approximately related by an isometry of the second kind, often an inversion. This mapping being only approximate, the corresponding operation is only a pseudo-symmetry operation, sometimes incorrectly called 'non-crystallographic' (Clevers & Coquerel, 2020) instead of partial or pseudo or pseudo-partial (Nespolo et al, 2008). This operation does not appear in the space group, allowing for a racemic compound in a Sohncke-type space group.…”

Nested Polyhedra and Indices of Orbits of Coxeter Groups of Non-Crystallographic Type

“…Considering the values appearing at each coordinate, the index I 2 of the subgroup G of H 3 is equal to 2 3 I 2 (H 3 ). Therefore, the embedding index γ = 3 2 .…”

“…A significant number of applications of non-crystallographic Coxeter groups in solid-state physics [1], chemistry [2] and structural genomics [3] motivates the current study. The symmetries of the H 2 and H 4 groups play an essential role in the construction and description of quasicrystals [4].…”