We know that taking the centers of the faces of any 3d polyhedron (say, the Platonic solids) produces the dual solid. And repeating this operation gives us back the original solid. Another possible thing we can do is take the centers of the edges. This will produce other solids as well. If you do this to a tetrahedron, you get an octahedron. Do this to an octahedron and you get a cuboctahedron. My question is, why does taking the face centers preserve the properties of the Platonic solids and have this nice dual-solid property while taking the edge centers doesn't. What makes the operation of taking the face centers "superior" to taking the edge centers? And is there a name for the process of forming a new solid by taking the edge centers?
It may be worthwhile visiting this MO question, The limit of edge-midpoint convex polyhedra, and in particular, the answer by A. Meyerowitz:
Image due to Aaron Meyerowitz.
The general setup here is dealing with regular polytopes. Then you are asking about taking the centers of k-faces (k-dimensional sub-elements). This process is usually called the k-rectification, cf. https://en.wikipedia.org/wiki/Rectification_(geometry).
When describing regular polytopes by Dynkin diagrams, those are of type xPoQoRoSo..., that is have a special node ("ringed", here demarked by x) at one end, and all other nodes are "unringed" (demarked by o), whereas the links bear several number marks (here represented by P, Q, R, etc.). The rectified polytope then gets described by oPxQoRoSo... and describes your edge-center figure, the birectified will be oPoQxRoSo... and describes the hull of the set of 2-boundary centers, etc.
What is special for the birectification, when applied to polyhedra, is that those would be oPoQx, that is the Special node happens to lye at the opposite end of the graph. And this - quite generaly - is just the dual of the starting regular polytope.
Whilst rectification (or multi-rectification like bi-rctified = oPoQxRoSo… etc.) is only defined on base of regular polytopes and which thus ensures all equal edges throughout (even when dealing with non-convex regulars), there is a closely related operation called ambification, which simply is defined by the convex hull of the edge centers of any convex polytope. That latter one has been used in the other answer. Applied to the convex regular ones those clearly coincide. But ambification generally will not ensure to produce equal edges in the new polytope.