Oh, I forgot to comment on animation. I wasn't confident enough in what you wanted to try animating it, but creating an animation is fairly easy. You just create a list of the individual "frames" and apply ListAnimate to it. Or you can parameterize how each frame is created, and use Animate. Maybe just rotating the plot (or even just rotating the ViewPoint) will get you what you want. Or maybe it's the Lissajous parameters themselves that you want to vary. Creating the frames you want will be the main challenge--animating them is a cinch.

Oh, sorry, I assumed too much.

If you're running Mathematica, you can just open a new notebook (File -> New -> Notebook). Copy the Lissajous3D code above, and paste it into the notebook. What you should have at that point is an Input cell. You'll see a bracket at the right side. With the cursor anywhere in that cell, tap Shift+Return. It won't produce any output, but it will have evaluated that code so that Lissajous3D is available to use like any other function. The cursor should now be below that cell. You can also just click anywhere in the whitespace below the cell to make sure. Now copy the whole ParametricPlot3D expression. Paste it into your Mathematica notebook. Again, with the cursor anywhere in this new cell, tap Shift+Return. This time it should output the plot, and it should look like the picture I pasted above. You can now tweak things in the ParametricPlot3D cell (don't need a new cell), and Shift+Return again.You can also tweak the Lissajous3D definition and Shift+Return that cell, and the next time you Shift+Return the ParametricPlot3D cell, it will use the updated definition.

Here's the message I've been trying to send for days but keeps giving the error. I finally figured it out. I believe it didn't like the emoji I had placed inline. Ugh!

Got it, thanks! I ended up finding a couple tutorials online that got me past the initial hurdles of "what in the world is a notebook?" (lol) and the secret of SHIFT-ENTER.

As to why the compiler didn't complain when I tried to execute the ParametricPlot3D code with the Lissajous3D definition missing, I have no idea :-)

Anyway, I think we can agree that this is officially "jumping into the deep end", and not the "Hello, World" first project that newbies would be tackling.

Meanwhile, I have an update. I've spent a while reverse engineering and playing with your code, documenting, and "manipulating" (pasted below and attached):

Lissajous3D[cx_, cy_, w_, d_, p_][t_, s_] := {cx Sin[t w + d], 
  cx Cos[t w + d], cy Sin[t] + Rescale[cx Cos[t w + d], {-1, 1},
    {s, p s}]}  

Manipulate[ParametricPlot3D[
   Lissajous3D[
    1, 1,                    (*overall aspect ratio *)

    w,                          (*vertical nodes*)
    \[Phi],        \
                   (*phase*)

    p]                            (*perspective: front/
   back ribbon thickness ratio *)
   [t, s],
        {s, 0, smax},    (*ribbon thickness *)
   {t, 0, 
   tmax},    (*period around cylinder*)
   ViewPoint -> Front,
   ViewProjection -> "Orthographic",
   ColorFunction -> (White &),
   MeshStyle -> None,
   PlotPoints -> {10, 60}, (*resolution??*)

  Lighting -> {{"Point", Blue, {0, -5, 0}}, {"Point", 
         Green, {-3, -5, 0}}},
   Axes -> ax]
 ,
 {{w,      1,         "vertical frequency"}, 1, 10, 
  Appearance -> "Labeled"},
 {{\[Phi],      Pi/4, "relative phase"}, 0, 2 Pi, 
  Appearance -> "Labeled"}, Delimiter ,
 {{smax, .1,      "ribbon thickness"}, .01, .3, 
  Appearance -> "Labeled"},
 {{tmax, 2 Pi,   "tmax"}, 0, 8 Pi, Appearance -> "Labeled"},
 {{p,     .25,      "perspective"}, 1, .1, Appearance -> "Labeled"},
 {{ax,   False, "show axes"}, {True, False}}
 ]

Before I get any further with graphical details, the main limitation is that this Lissajous function only allows control over the vertical frequency, not the horizontal frequency. Specifically, the "t" for the two parametric functions needs to be independent. I tried splitting that out (code not shown, because it doesn't work), but things don't look right.

Do you know how to fix that without breaking the 3D features we have here?

Thanks again!

Dave

p.s. Happy New Year!

Attachments:

lissajous.nb

As to why the compiler didn't complain when I tried to execute the ParametricPlot3D code with the Lissajous3D definition missing, I have no idea :-)

Thought it might be helpful to comment on this. There isn't really a compiler in the common sense, and there is nothing directly analogous to static type checking. ParametricPlot3D was given two parameters and their ranges. It was given a "function" (an expression that looked like what you might think of as a function) that was being applied to those two parameters. That function (Lissajous3D), however, didn't have any rewrite/evaluation rules associated with it (because you hadn't evaluated the SetDelayed expression that "defined" the Lissajous3D "function"). So, ParametricPlot3D was just left with a bunch of stuff that it didn't know how to interpret as a plot, and so the resulting output just looked empty.