Hello, is it so that symmetries on the plane are essentially discrete subgroups of the group of isometries on the plane? If that is true, then why should we think that the only symmetries in the plane are given by the wallpaper group and the point group? Can't we just change the metric of the 2D plane and obtain new kind of symmetries? For instance, could we just consider the space ℝ1,1 with norm ||p||2=x2-y2, where p=(x,y), and find new subgroups of isometries w.r.t. this metric?