Hexagonal projected symmetries

Abstract

Abstract. In the study of pattern formation in symmetric physical systems a 3-dimensional structure in thin domains is often modelled as 2-dimensional one. We are concerned with functions in R 3 that are invariant under the action of a crystallographic group and the symmetries of their projections into a function defined on a plane. We obtain a list of the crystallographic groups for which the projected functions have a hexagonal lattice of periods. The proof is constructive and the result may be used in the study of observed patterns in thin domains, whose symmetries are not expected in 2-dimensional models, like the black-eye pattern.

Projections of patterns and mode interactions

“…The group Γ L = R n L + J is contained in Γ L, but does not necessarily coincide with it, since the inclusion J ⊂ H L may be strict. This is the case in Example 2 above, other examples appear in [19]. Theorem 9.…”

“…Information about which symmetries are preserved under projection has been obtained by Labouriau and Pinho in [17,21]. In Oliveira et al [19], we describe which symmetries can lead to a projected function with hexagonal symmetry. In studying the dynamics of (n + 1)-dimensional patterns by observing their n-dimensional projection, it is desirable to extract information from the symmetries that can be seen in the n-dimensional space directly.…”

Periodic Functions, Lattices and Their Projections

“…For most lattices, only some special projection widths yield a hexagonal lattice. The cases where for all the projection widths there is a three-fold rotation in the group of Π y0 (X Γ ) have been discussed in [14]. These correspond to the primitive, the body-centered and face-centered cubic lattices, the rhombohedral lattice and the hexagonal lattice in 3 dimensions, always in special positions.…”

“…In these special cases, projection always yields a hexagonal lattice, but the size of the cell varies. The triclinic lattices in case 1 do not appear on the list in [14] because their symmetry group does not contain any rotation. Generically, the holohedry of a lattice in case 1 is trivial.…”

“…Of these, only four types yield a hexagonally periodic projection for all widths of the projecting band. Three of these have been reported in [14]. Patterns for the fourth type are provided in Section 8, along with a discussion of these examples.…”