The inverse of the mapping:
R^3 --> R^3 : p --> 2 p / ( 1- p²)
maps the plane at infinity onto the unit sphere. We also end up with two copies of the euclidean space: One inside the unit sphere and the other outside.
In these pictures, the Barth sextic is transformed by the above mentionned mapping (After some simplification we obtain an algebraic equation. The cool thing is that the dodecahedral symmetry is preserved). In the first picture the outer copy is colored in purple.

Here are the Fragmentarium shaders :
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