Parameterise linear combination of cosines How do I parameterise the following implicit surface?
$$ \cos x + \cos y + \cos z = 0 $$
Motivation for this problem comes from attempting to find stable motion for an object balanced on one point. The equation seems so simple, but I really have no idea how to solve this problem. Apparently the surface approximates the Schwarz P minimal surface, however I do not have any knowledge of the theory behind that.
 A: The first octant of the "Fermi surface" can be parametrized as
$$
\begin{pmatrix} x \\ y \\ z \end{pmatrix} =
\begin{pmatrix}
  \cos^{-1}
  \left[
    -\dfrac{u}{2}+v\left( 1-\dfrac{|u|}{2} \right)
  \right] \\[5pt]
  \cos^{-1}
  \left[ 
    -\dfrac{u}{2}-v\left( 1-\dfrac{|u|}{2} \right)
  \right] \\[5pt]
  \cos^{-1} u
\end{pmatrix}$$
where $u$, $v\in [-1,1]$.
To make it periodic,
$$\begin{pmatrix} x \\ y \\ z \end{pmatrix} =
\begin{pmatrix}
  2\pi \left \lfloor \dfrac{i+1}{2} \right \rfloor+(-1)^{i}
  \cos^{-1} \left[ -\dfrac{u}{2}+v\left( 1-\dfrac{|u|}{2} \right) \right] \\[5pt]
  2\pi \left \lfloor \dfrac{j+1}{2} \right \rfloor+(-1)^{j}
  \cos^{-1} \left[ -\dfrac{u}{2}-v\left( 1-\dfrac{|u|}{2} \right) \right] \\[5pt]
  2\pi \left \lfloor \dfrac{k+1}{2} \right \rfloor+(-1)^{k}\cos^{-1} u
\end{pmatrix}$$
where $i$, $j$, $k\in \mathbb{Z}$.
For first "Brillouin zone", take $i$, $j$, $k=-1$, $0$.
A: Just to get an idea of the surface:

The curves of different colour correspond to different values of $\cos z$
with $z = k \pi/4$.
