3D tiling patterns with the symmetry of the extended Picard group are automatically generated by means of dynamic systems equivalent to the extended Picard group.The derived patterns have the symmetry of the wallpaper group pmm on the cross section plane parallel to the xy plane

and own the symmetry of the extended modular group on the cross section plane containing the x axis.The numerical experiments show that the generated patterns are artistic.The method provides a novel approach to devise exotic symmetric tiling patterns from a dynamic system’s point of view.