Ordinarily, Lissajous curves are confined to oscillations in 2D (see: Lissajous Curves), but it is possible to add a z equation to make the curve oscillate in 3D. Here are the parametric equations:
x = a * sin(pt)
y = b * sin(qt)
z = c * cos(rt)
There are many possibilities in how you arrange the different variables, recognizing that a,b and c influence the height in any particular direction, and p,q, and r influence how fast it oscillates in that direction. You can also change any one of the sines to cosine, which will change the phase in one dimension (or you can experiment with adding your own phase changes by using equations sin(pt+n)).
From the top corner, going clockwise, the numbers I used (in mm) are:
x = 50 * sin(13t)
y = 50 * sin(14t)
z = 50 * cos(15t)
x = 150 * sin(23t)
y = 10 * sin(5t)
z = 150 * cos(3t)
x = 150 * sin(at)
y = 150 * sin(bt)
z = 150 * cos(bt)
x = 50 * sin(6t)
y = 50 * sin(25t)
z = 50 * cos(13t)
x = 50 * sin(6t)
y = 50 * sin(10t)
z = 50 * cos(13t)
x = 150 * sin(6t)
y = 150 * sin(13t)
z = 150 * cos(50t)
These could make nice hot plates, lamps, bracelets, or just cool decorations. Check out the Personal Pages of Robert Ferreol for more background and mathematical applications.
If you want to try this yourself, the Grasshopper code is below, or you can try typing these equations into a 3D parametric plotter online.
y = 50 * sin(14t)
z = 50 * cos(15t)
y = 50 * sin(25t)
z = 50 * cos(13t)