Groups and Symmetry

On the projection of functions invariant under the action of a crystallographic group

“…The reader is referred to Armstrong [1] for results on Euclidean and plane crystallographic groups, to Senechal [17] and Miller [14] for results on lattices and crystallographic groups. A detailed description may be found in Pinho [16].…”

“…Therefore, case (2) is also impossible. For case (3) we follow the arguments of case (1). As L * = N * y0 ∪ M * + ∪ P * then Q * ∩ N * y0 ∪ M * + is an infinite set and at least one of the periods τ y0 or τ + must exist.…”

Notiz: Quantification of Symmetry

“…The definition of sq is based on the notion that both the number and order of symmetry elements should be >D 2h (15)>D 3 (13)>C 4 (ll)>D 2 (7) = C 3 (7) > C 2 (3) considered when deciding overall symmetry ranking for a particular group. The order of the individual symmetry operator (element) gives the number of symmetrically equivalent positions of an object that can be reached after applying symmetry transformation associated with this element.…”

“…In order to address this question on a more general level one must turn to group theoretical considerations because, as Armstrong [3] has pointed out, "numbers measure size, groups measure symmetry".…”

Kodkod: A Relational Model Finder

“…Because every relational constant is isomorphic to a graph, Thm. 1 equates the task of finding Sym(P ) to that of computing the automorphisms of the graphs that correspond to the constants in D-a problem with no known polynomial 1 Recall that cycle notation for permutations [34] indicates that each element in a pair of parenthesis is mapped to the one following it, with the last element being mapped to the first. The elements that are fixed under a permutation are not mentioned, i.e.…”