We consider n-dimensional Euclidean lattice networks with nearest neighbor coupling architecture. The associated lattice dynamical systems are infinite systems of ordinary differential equations, the cells, indexed by the points in the lattice. A pattern of synchrony is a finite-dimensional flowinvariant subspace for all lattice dynamical systems with the given network architecture. These subspaces correspond to a classification of the cells into k classes, or colors, and are described by a local coloring rule, named balanced coloring. Previous results with planar lattices show that patterns of synchrony can exhibit several behaviors such as periodicity. Considering sufficiently extensive couplings, spatial periodicity appears for all the balanced colorings with k colors. However, there is not a direct way of relating the local coloring rule and the coloring of the whole lattice network. Given an n-dimensional lattice network with nearest neighbor coupling architecture, and a local coloring rule with k colors, we state a necessary and sufficient condition for the existence of a spatially periodic pattern of synchrony. This condition involves finite coupled cell networks, whose couplings are bidirectional and whose cells are colored according to the given rule. As an intermediate step, we obtain the proportion of the cells for each color, for the lattice network and any finite bidirectional network with the same balanced coloring. A crucial tool in obtaining our results is a classical theorem of graph theory concerning the factorization of even degree regular graphs, a class of graphs where lattice networks are included.

Abstract. We study functions defined in (n + 1)-dimensional domains that are invariant under the action of a crystallographic group. We give a complete description of the symmetries that remain after projection into an ndimensional subspace and compare it to similar results for the restriction to a subspace. We use the Fourier expansion of invariant functions and the action of the crystallographic group on the space of Fourier coefficients. Intermediate results relate symmetry groups to the dual of the lattice of periods.

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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.

A Ab bs st tr ra ac ct t. . The perceivable regularity of some Roman villas can be understood in the context of a grid-based design. In this paper we try to clarify the requirements under which we may consider that a villa has an outline based on a grid and we quantify the accuracy of the correspondence between a villa's plan and a given grid. We follow this approach with some Roman villas in Portugal and use the grids as tools for the analysis and the reconstruction of their plans.

Consider a periodic function f of two variables with symmetry Γ and let L ⊂ Γ be the subgroup of translations. The Fourier expansion of a periodic function is a sum over L * , the dual of the set L of all the periods of f. After projecting f , some of its original symmetry remains. We describe the symmetries of the projected function, starting from Γ and from the structure of L * .

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