[math-fun] Lissajous illusion
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
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
This is at least as much illusion as geometrical coincidence. The paradoxical shape here is the left one, which is tumbling in the vertical plane but appears to be rotating in the horizontal. So an intriguing exhibit would be to tip its rotation axis upright and hang it from the ceiling, just above eye level, on an inconspicuous, twisting wire. Then it ought to appear tumbling in the same vertical plane as any sufficiently distant observer (inviting disputes). Sadly, this works only marginally. The "eye" is much more receptive to horizontal movements than vertical, so the true rotation axis is much more apparent in the hanging configuration. If you have a laptop, turn the lissajoke.gif animation sideways. But this is cool, too--an object which insists on rotating in the horizontal plane when you try to tilt it vertical. This horizon bias may somehow relate to our eyes being side by side, although closing one eye here is clearly a no-op. But I wonder if a congenitally one-eyed observer has the same perceptual bias. Probably. We all walk upright on a flat planet. But why do fish bother to swim upright? Anyway, Neil has made a beautiful animation of the figure rotating about the axis parallel to the intersection of the floor and opposite wall: http://gosper.org/lissagent.wmv and http://gosper.org/lissajous.gif . In OS-X, I was unable to view the animated gif beside a (hanging) version rotated 90°, but adjacent wmv players work great [Edit/Rotate Left][View/Loop]. George, can this really be new? I recall a helicoidal mylar hanging decoration that appears to coruscate vertically when twisted, but it's nowhere near as convincing. --rwg On Wed, Feb 6, 2013 at 12:39 AM, Bill Gosper <billgosper@gmail.com> wrote:
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
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Bill Gosper