We find periodic skyrmionic textures via conformal cartographic projections that map either an entire spherical parameter space or a hemisphere onto every regular polygon that provides regular tessellations of the plane. These maps preserve the sign of the Skyrme density throughout the entire space. We implement these textures in the polarization state of a laser beam, and demonstrate that paraxial fields where a periodic texture preserving the sign of the Skyrme density is implemented in the polarization state distribution unavoidably exhibit zeros.