Turn 2D Mesh into a 3D Mesh reuleaux solids

I am trying (right now failing) to construct a 3D reuleaux solid in blender. For those of you who don't know a reuleaux solid basically is a fancy word for solids of constant width. If you are a math nerd like me then you should go check them out as they are quite fascinating! (Here is a link to a informative Wikipedia page: Reuleaux Tetrahedron Wikipedia

Right now I have the 2D version of it and it looks like this:

If anyone is wondering how I constructed this I accomplished it like this: Its basically the same way you would construct an equilateral triangle in geometry class.

My question is how can I take this 2D mesh object and create a 3D mesh object such as this Reuleaux tetrahedron:

I tried using the screw modifier but the blender gods laughed in my face. I think that there must be some way of rotating the figure around itself to accomplish this but I have yet to come upon that solution.

Thanks for the help!

The wikipedia article says

Four balls intersect to form a Reuleaux tetrahedron.

Which is easy to implement with boolean operations (and balls).

5. The distance of the centers of the spheres is the length of an edge of the pyramid. The radius of the spheres should equal this value. (The dimensions (diameters in this case) will naturally be twice the value.) In my case the pyramid has an edge length of 1.633, hence the spheres have to be rescaled to match a dimension of 1.633 * 2.