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Triply periodic surfaces

  1. Feb 7, 2014 #1
    So I understand that a surface is triply periodic when the surface is invariant under three tanslations in [itex]R^{3}[/itex]. When looking at the primitive for example, how is that translation defined? Say that the primitive is a set defined by the equation

    [itex]cos(x)+cos(y)+cos(z)=0[/itex]

    My guess is that the translation would take [itex](x,y,z)\rightarrow(x+\Delta x, y+\Delta y, z +\Delta z)[/itex]. Is that incorrect thinking?
     
  2. jcsd
  3. Feb 11, 2014 #2
    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

    [itex]\nabla(cos(x)+cos(y)+cos(z)) \ne 0 \rightarrow (-sin(x),-sin(y),-sin(z)) \ne 0 \rightarrow x,y,z \ne 0[/itex]

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