It would be worthwhile for the reader to reflect on the relationship between the relations and the syzygies of and singularities of plane curves. One such codimension one singularity is the triple point , which corresponds to the last two relations above, which can be viewed as ``the motion of a double point across a line''. One such codimension two singularity is the quadruple point , and it corresponds to the syzygy of Figure 1: There is a circle-worth of generic deformations of the quadruple point, corresponding to ``the cross rotating around the target'': . The different codimension one singularities along this rotation are exactly the relations in our syzygy.