these are a generalisation of the cupola and bicupolic rings (which are lever 1 and 2 respectively:D)
I've only just started studying these, so I can't say much atm
the simplex and hypercube cases can be CRF polytopes but only the simplex, square, and cube forms can be gyrated (with the latter the cube becomes a octahedron. (btw I haven't yet ennumerated gyrations and dimishes of the general case case)
this because a regular polytope needs to have a ditope angle (angle between the facets) of less than 120 degrees in order to form a cupola (the 16 cell is exactly 120 so doesn't work._
but I'm not going to restrict my self to CRF, this is a seperate investigation:D
to start with there's an interesting property of the hypercube cases (which is what I'm using as a foundation to actually define these shapes) (the simplex also have similar properties that I haven't looked at properly yet)
3D
Level one: if you take a rhombicuboctahedron it can be cut into 3 pieces:
square cupola + 8-prism + square cupola
4D
Level one: if you take a runcinated tesseract it can be cut into 3 pieces
cube cupola + rhombicuboctahedron prism + cube cupola
Level two: if you take a cube cupola it can be cut into 3 pieces
square bicupolic ring + square cupola prism + square bicupolic ring
(btw it's "orthobi":D)
5D
Level one: if you take a stericated penteract it can be cut into 3 pieces
tesseract cupola + runcinated tesseract prism + tesseract cupola
Level two: if you take a tesseract cupola it can be cut into 3 pieces
cube bicupolic ring + cube cupola prism + cube bicupolic ring
(btw neither cube has been changed to an octahedron)
Level three: if you take a cube bicupolic ring it can be cut into 3 pieces
square tricupolic screw-guage + square bicupolic ring prism + square tricupolic screw-guage
6D
Level one: if you take a pentalated hexeract it can be cut into 3 pieces
penteract cupola + stericated penteract prism + penteract cupola
Level two: if you take a penteract cupola it can be cut into 3 pieces
tesseract bicupolic ring + tesseract cupola prism + tesseract bicupolic ring
Level three: if you take a tesseract bicupolic ring it can be cut into 3 pieces
cube tricupolic screw-guage + cube bicupolic ring prism + cube tricupolic screw-guage
Level four: if you take a cube tricupolic screw-guage it can be cut into 3 pieces
square quadricupolic screw-guage + square tricupolic screw-guage prism + square quadricupolic screw-guage