Symmetries of projected wallpaper patterns

Abstract

Abstract. In this paper we study periodic functions of one and two variables that are invariant under a subgroup of the Euclidean group. Starting with a function defined on the plane we obtain a function of one variable by two methods: we project the values of the function on a strip into its edge, by integrating along the width; and we restrict the function to a line. If the functions had been obtained by solving a partial differential equation equivariant under the Euclidean group, how do their symmetries compare to those of solutions of equations formulated directly in one dimension?Some of the symmetries of projected and of restricted functions can be obtained knowing the symmetries of the original functions only. There are also some extra symmetries arising for special widths of the strip and for some special positions of the line used for restriction. We obtain a general description of the two types of symmetries and discuss how they arise in the wallpaper groups (crystalographic groups of the plane). We show that the projections and restrictions of solutions of p.d.e.s in the plane may have symmetry groups larger than those of solutions of problems formulated in one dimension.

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

“…Thus, there exists the period τ y0 implying that Q * − N * y0 is either the empty set or an infinite set. Since Q * − N * y0 ⊂ (Q * ∩ P * ) is finite, it follows that L * = N * y0 , which implies that (0, y 0 ) ∈ L. However, by the statement (12), under this condition, M * y0 cannot be a subset of P * and so case (1) cannot occur.…”

“…At first, we prove the statement: (12) If (0, a) ∈ L for some a = 0 then M * y0 ⊂ P * . If (0, a) ∈ L for some a = 0 then property (2) of the bases, in Lemma 5.2, ensures that M * y0 has n linearly independent generators, l * i = (a * i , 0) for i ∈ {1, .…”

“…The result only depends on the formal expansion, and also holds under less restrictive hypotheses on X Γ , given in Section 2 below. The particular case of a projection from the plane into a line has been treated in [12], where a different proof of Theorem 1.2 is given for the case n = 1. In this dimension, the only symmetries that remain after projection are translations and reflections, which simplifies the analysis.…”

Projections of patterns and mode interactions

“…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.…”

Periodic Functions, Lattices and Their Projections

“…When we project periodic functions we may obtain functions with too few periods. For planar lattices and wallpaper patterns this follows from the results of [10,11], here we treat the general case.…”