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 $f$ defined on the plane we obtain a function of one variable by two methods: (1) we project the values on a strip into its edge, by integrating $f$ along the width; and (2) we restrict $f$ to a line. If the functions had been obtained by solving a partial differential equation equivariant under the Euclidean group, how do the symmetries of the function of one variable compare to those of solutions of equations formulated directly in one dimension?

Some of the symmetries of projected and of restricted functions may be obtained knowing only the symmetries of the original functions. Extra symmetries arise at special widths of the strip and at 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 (crystallographic groups of the plane). We show that the projections and restrictions of solutions of differential equations in the plane may have symmetry groups larger than those of solutions of problems formulated in one dimension.