Thoughts on Drawing a 3-D Star

I don't really know where I am going with this, but I think it is kind of interesting. The question was: "How do you draw a 3-d star?" I could not figure out how NOT to end up with a star which had more than 5 points. There was one way, but it was not quite right. The not-quite-right way was to take this blue-green star (which I filled in with color and on which I drew these black lines and numbers) and to take another "3-d" star just like it and glue it on the back of it so it was a fatter version of what you see here, But it would not be right. It would still be too FLAT. So then I thought about what we'd have to do to make it "spherically fat." This lead me to the idea of taking THE BASES of each of the 5 3-dimensional pyramids and arranging them on such a sphere. The center points (of the bases) would have to end up being equidistant from the three nearest base point centers. This rule would have to apply to ALL points whose 3 neighbors "fit that picture." I tried to envision what this would do. The image of what would occur is not yet what we are looking for though. I think I know what would have to be done as far as physically moving and re-designing the star tips. But the math might be wicked hard to do. There would be a similar difficulty in drawing the new image, since we would have to consider the drawing in light of star points we could not actually see (but which we would need to know (if only in faint, dotted-line format). If you are still reading this, I think what would need to be done is to WIDEN the 3 angles (equally) for the interior star tips so that the ENDS of all the star point edge lines would EACH end up at one point in the converging valley of respective pairs of adjacent star point BASES. (And if you understood that, your may be highly brilliant). I have to confess I am about as clueless on how to draw this as my explanation may be accurate. One thing I am curious about is IF one of the base converging points will end up in the center of the star we see here. THAT POINT would have show up in 4 other points of equal separation on the sphere. Also, I'm pretty sure the star we have in our minds would NOT have any visible curved surfaces or exposed "sphere-only" surfaces (or would it)? Now I am beginning to think it might have to. Maybe those areas would be ellipses with pointed ends (or convex lens shapes). The biggest question I have is how the heck does one find 5 points of equal separation distance on the face of a sphere. I began by looking at the bases of the 5 pyramids, then thinking of "stretching them" so they all fit nicely around the sphere, but this whole process involves a COMPLETE SHIFT of the x-y-z axis system(s) for the star-tips. Sorry for all this thinking out loud. It may be hard to understand. It is for me. Maybe someone brilliant out there can actually answer some of these questions. Better yet, maybe they could design such a star.