Because It's There

Same basic idea as Phoenix, this time with the top half of the surface $|x|+y^2+\sqrt[3]{z}=1$.

By the way, I'm now selling prints if you're interested in putting some of my work on your wall.

Here's the code:

DynamicModule[{cols},
 cols = RGBColor /@ {"#A7CDCC", "#F87D09", "#A7CDCC", "#004A55"};
 Manipulate[
  Show[Table[{ParametricPlot3D[{{(1 - z^(1/3) - y^2), y, 
        z}, {-(1 - z^(1/3) - y^2), y, z}}, {z, 0, 
       Max[(1 - y^2)^3, .001]}, PlotRange -> 2, Boxed -> False, 
      Axes -> None, ViewPoint -> 65 {1.2, 1.5, 1.7}, 
      ViewCenter -> {.504, .5, .55}, ViewAngle -> ?/1200., 
      ImageSize -> 540, 
      PlotStyle -> 
       Directive[Blend[cols[[;; 3]], (y + 1)/2], Thickness[.004]], 
      Background -> cols[[4]]], 
     Graphics3D[{Thickness[.004], Blend[cols[[;; 3]], (y + 1)/2], 
       Line[{{# (1 - y^2), y, 0}, {2 #, y, 0}}] & /@ {-1, 
         1}}]}, {y, -2 + t, 2, 1/8}]], {t, 0, 1/8}]]