This simple fact might make a nice math objet d'art. Labeled[Manipulate[ Labeled[ParametricPlot[{{t - 2*\[Pi] - 1, Cos[a*(t + v) + \[Phi]]}, {Sin[b*(t + v)], t - 2*\[Pi] - 1}, {Cos[t], Sin[t]}/11 + {Sin[b*v], Cos[a*v + \[Phi]]}, {Sin[b*v], t/\[Pi] - 1}, {t/\[Pi] - 1, Cos[a*v + \[Phi]]}, {Sin[b*t], Cos[a*t + \[Phi]]}}, {t, 0, 2*\[Pi]}, PlotRange -> {{-4 - \[Pi], 9/8}, {-4 - \[Pi], 9/8}}], {StringJoin[ "Vertical frequency a=", ToString[a]], StringJoin["Horizontal frequency b=", ToString[b]], StringJoin["Phase \[Phi]=", ToString[\[Phi]/\[Pi]], "\[Pi]"]}, {Top, Right, Bottom}, RotateLabel -> True], {v, -\[Pi], \[Pi]}, {\[Phi], 0, 2 \[Pi]}, {a, Range[5]}, {b, Range[5]}], "Lissajous Mechanism", Top] Click frequencies a to 3, b to 4, say. Pop open the + sign on the phase (φ) parameter and run it slowly. Imagine the precessing Lissajous figure to be drawn on a rotating transparent cylinder. Is it vertical or horizontal? So make two metal "Lissajous hoops", one for each axis and rotate them side by side on mutually perpendicular turntables. --rwg