How to make a tetrahedron solid and grid? I'm working on a build-up animation. See images. All the balls and cylinders have to follow a global equilateral triangle grid. But I don't have this grid and the result is not precise, it doesn't fit. My best solution would be to make a perfect equilateral triangle solid (4 equal faces - 3 verts each). But I can't find such a primitive in Blender, even in the extra objects add-on. Can someone help me out, how to build this solid, so I can replicate it further and make myself a 3D grid from it? Or maybe there is a clever way to make a grid like this directly, something I can snap to?

Ok. I got a "perfect" tetrahedron thanks to Grimm. But it is not perfect. I began making a grid, see 3rd image. But it is inaccurate. It doesn't fit. I need a perfect grid. Somebody knows how to achieve a perfect tetrahedron grid? Maybe there is a script somewhere that does it? All ideas welcome.
This is fundamentally impossible because (surprisingly!) the platonic tetrahedron is not space-filling.
The dihedral angle of two adjacent faces of a regular tetrahedron is arccos(1/3) ≈ 70.53. So if you put 5 regular tetrahedrons together it makes 70.53*5=352.65 which is not quite the 360 you would need to fill space.