In many examples of repeating patterns in the art of various cultures, the use of symmetries to analyze those patterns does a good job of capturing the repetition intended by the artist. In other cases, however, the artist uses precise forms of repetition that are not well modeled by mathematical symmetries. The analysis of the orbits of motifs under the action of the symmetry group both reveals situations where this happens, and gives us direction as to what else is needed to model the artists' apparent intent. The ways in which the art of different cultures need different types of extensions of symmetry ideas reveals structural differences in the design art of those cultures, and hence in their ethnomathematics.