Understanding Triply Periodic Surfaces: Translations and Invariance?

[Demon117]

So I understand that a surface is triply periodic when the surface is invariant under three tanslations in R^{3}. When looking at the primitive for example, how is that translation defined? Say that the primitive is a set defined by the equation

cos(x)+cos(y)+cos(z)=0

My guess is that the translation would take (x,y,z)\rightarrow(x+\Delta x, y+\Delta y, z +\Delta z). Is that incorrect thinking?

 

[Demon117]

Readjustment

So I have new information now. Apparently showing that the gradient of a level set does not vanish somehow also shows that a set defined as above is invariant under three translations. How is that the case?

With that in mind, the gradient of the above is

\nabla(cos(x)+cos(y)+cos(z)) \ne 0 \rightarrow (-sin(x),-sin(y),-sin(z)) \ne 0 \rightarrow x,y,z \ne 0

I don't see how this guarantees the surface is invariant under three translations in R^{3}. Any suggestions?