Building the model is probably the best way to go, even if mentally, so you may need to visualise quite strongly; er .. and consider it within an ‘xyz ‘ environment.
Using a circular cookie/biscuit/scone cutter [or something of that nature] cut a cylindrical form that is longer than the cylinder’s diameter.
At 90 Deg to that cylinder’s axis, cut another cylinder. A third cut is now needed in the 3rd dimension.
A cylinder has been created and trimmed in the 2nd and 3rd dimensions (or elevations).
The object has the initial appearance of a sphere but with a subtle resemblance to a cube.
The object has 8 vertices (as does the cube). [IMPORTANT - See NOTES below].
All surfaces are curved cylindrically. There are 12 of these. Each dimension has 4. (On a cube there are 12 edges).
The ‘X’s have their endpoints at the vertices. There are 6 of these (the cube has 6 faces). Between adjoining surfaces the edges are sharp at the vertex but fade into oblivion at the midpoint of each ‘X’.
The vertices remain the same but the values of edges and faces are inversed.
QUESTION ( as an aside, but a thought): Disregarding the ‘Plan/Elevation’ stipulation of this submission, what forms do: eg, a tetrahedron, dodechahedron, … take if we incur a similar inversion process?
NOTES from Charlie. He has passed me these notes which I accept. Note Vertices.
Actually the object has 14 vertices: 8 where 3 of the surfaces meet, corresponding to the vertices of the cube, and 6 where 4 surfaces meet, corresponding to the faces of the cube.
The cylindrically curved faces correspond to the edges of the cube, and if flattened out would form a rhombic dodecahedron.
The extra 6 vertices are the midpoints of the X's, and though the edges "fade into oblivion" there, it's still a vertex. Like the apex of a vaulted ceiling where perpendicular cylindrical surfaces meet.