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The surface is just a union of straight lines, known as a ruled surface. It also looks good, especially in translucent material with some low amount of infill (see picture). The openSCAD code is also included and you can find it on github too: https://github.com/tom111/PiIstGenau3D
Mathematically, the Segre embedding is a map of a product of projective spaces into a higher dimensional projective space, and this is its non-negative real part.