I have a definition of a regular tetrahedron by the distance of each plane from the origin.
If the distance is negative, the plane is on the far (negative Z) side of the origin.
The four planes are named after the four cartesian quadrants: i, ii, iii, and iv, in a right-hand rule coordinate system,
such that positive i places the plane up (positive Y), to the right (positive X), and closer (positive Z).
All positive values define planes with positive Z-axis intercepts.
In the image, the green face is i, ii is red, iii is blue, and iv is opposite (not shown).
So for a tetrahedron with i=sqrt(3), ii=0, iii=0, and iv=0
the vertices (in no particular order) would be (0,0,0), (1,1,0), (0,1,1), (1,0,1).
I feel like I should be able to solve this. I'm not sure what's blocking me.