The two "physobs" in http://gosper.org/lissajoke.gif are rigidly rotating about perpendicular axes at different rpm. In the "fish phase", the silhouette is an arc of a semicubical parabola 2 y^2 == (1 + x) (-1 + 2 x)^2 . (Sedately) kinetic sculpture? --rwg (with crucial help from Julian) Manipulate[ Show[ParametricPlot3D[ Evaluate[{{-Sin[2*t], Cos[3*t], Sin[3*t]}.{{1, 0, 0}, {0, Cos[+\[Phi]/2], -Sin[+\[Phi]/2]}, {0, Sin[+\[Phi]/2], Cos[+\[Phi]/2]}}, {5/2, 0, 0} + {Sin[2*t], Cos[3*t], Cos[2*t]}.{{Cos[+\[Phi]/3], 0, Sin[+\[Phi]/3]}, {0, 1, 0}, {-Sin[+\[Phi]/3], 0, Cos[+\[Phi]/3]}}}], {t, 0, 2*\[Pi]}, PlotStyle -> Directive[(*Opacity[0.7],*)CapForm[None], JoinForm["Miter"](*, Red*)], PlotRange -> All, ColorFunction -> (Hue[6 #4] &), Boxed -> False, MaxRecursion -> 0, PlotPoints -> 100, Axes -> None, Method -> {"TubePoints" -> 30}, ViewPoint -> {x, y, z}] /. Line[pts_, rest___] :> Tube[pts, 0.05, rest], Graphics3D[{Opacity[.1], Polygon[Table[ RotationTransform[-\[Phi]/3, {1, 0, 0}]@{{-1, Re[#], Im[#]} &[ 2*Exp[I*\[Pi]*(2*k + 0/2)/3]], {1, Re[#], Im[#]} &[ 2*Exp[I*\[Pi]*(2*k + 0/2)/3]], {1, Re[#], Im[#]} &[ 2*Exp[I*\[Pi]*(2*k + 4/2)/3]], {-1, Re[#], Im[#]} &[ 2*Exp[I*\[Pi]*(2*k + 4/2)/3]]}, {k, 3}]], Polygon[ Table[TranslationTransform[{5/2, 0, 0}]@ RotationTransform[-\[Phi]/2, {0, 1, 0}]@{{-1, Re[#], Im[#]} &[ 2*Exp[I*\[Pi]*(2*k + 0/2)/3]], {1, Re[#], Im[#]} &[ 2*Exp[I*\[Pi]*(2*k + 0/2)/3]], {1, Re[#], Im[#]} &[ 2*Exp[I*\[Pi]*(2*k + 4/2)/3]], {-1, Re[#], Im[#]} &[ 2*Exp[I*\[Pi]*(2*k + 4/2)/3]]}, {k, 3}]], Opacity[0], Line[{{-4/2, -3, -3}, {9/2, 3, 3}}]}]], {\[Phi], 0, 2*\[Pi], Appearance -> "Open"}, {x, {0, 1, 9, 69}}, {y, {0, 1, 9, 69}}, {{z, 69}, {0, 1, 9, 69}}] On Wed, Jan 30, 2013 at 9:25 PM, Bill Gosper <billgosper@gmail.com> wrote:
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