Can I inscribe a regular tetrahedron to a torus? What kind of regular tetrahedra can we inscribe into a given torus?
For example, is it possible to inscribe a unit edge length regular tetrahedron to a spindle torus with major radius $R=\frac{\sqrt 2}2$ and minor radius $r=\frac{\sqrt 3}2$?
It seems to me that such an inscription should be doable for any $R$ and $r$ in a reasonable range, but I'm having a very hard time imagining it; I hope someone with some knowledge/vision in spatial geometry can answer this from the top of their head.
 A: One solution has the vertices:
$$\begin{array}{ccc}
 (-0.42, & \ \ \, 0.28, & 0.84 )\\
 (-0.03, & -0.16, & 0.04 )\\
 (-0.19, & -0.69, & 0.87 )\\
 (\ \ \, 0.54, & \ \ \, 0.00, & 0.85 )\\
\end{array}$$
which lead to this view looking up at the region with $z>0$:

This comes from Mathematica using NMinimize, which surprisingly worked better than NSolve or FindInstance.  The code is below, and you could get other solutions by rotating this or by adding a term like $(a_2-\frac32)^2$ to the sum of squares.
xyz[a_, b_] := {(Sqrt[2] + Sqrt[3] Cos[b]) Cos[a],
     (Sqrt[2] + Sqrt[3] Cos[b]) Sin[a], Sqrt[3] Sin[b]} / 2

d[t_, u_, v_, w_] := (xyz[t, u] - xyz[v, w]).(xyz[t, u] - xyz[v, w])

sol = NMinimize[
     (d[a1, b1, a2, b2] - 1)^2 + (d[a2, b2, a3, b3] - 1)^2 + 
     (d[a3, b3, a1, b1] - 1)^2 + (d[a1, b1, 0, b0] - 1)^2 +
     (d[a2, b2, 0, b0] - 1)^2 + (d[a3, b3, 0, b0] - 1)^2,
     {a1, a2, a3, b1, b2, b3, b0}][[2]]

Round[{xyz[a1, b1], xyz[a2, b2], xyz[a3, b3], xyz[0, b0]} /. sol, .01]

tetra = ListPlot3D[{xyz[a1, b1], xyz[a2, b2], xyz[a3, b3], 
     xyz[0, b0]} /. sol, PlotTheme -> Business, 
     PlotRange -> {{-2, 2}, {-2, 2}, {0, 1}}]

torus = ParametricPlot3D[xyz[a, b], {a, 0, 2 Pi}, {b, 0, 2 Pi}, 
     PlotRange -> {{-2, 2}, {-2, 2}, {0, 1}}]

Show[tetra, torus]

