Just started work on the first problem set. It asks about a regular tetrahedron, and puts hydrogen atoms at the vertices and a carbon atom in the center. However, I'm forgotten what the center of a tetrahedron actually is, mathematically?

Question

Just started work on the first problem set.  It asks about a regular tetrahedron, and puts hydrogen atoms at the vertices and a carbon atom in the center.  However, I'm forgotten what the center of a tetrahedron actually is, mathematically?

phi · Answer

based on the symmetry we should expect the carbon atom to be equidistant from each hydrogen atom.  We start with a cube whose center is the origin.  We know we can circumscribe a sphere about the cube, so we know the vertices of the cube are all equidistant from the circle's  (and cube's) center i.e. the origin.

The vertices of the tetrahedron embedded in the cube are also vertices of the cube.  Therefore the point in the tetrahedron equidistant from each of its vertices is the origin (center of the cube).