I am trying to understand more about combinatorics and geometry of polytopes. If we create a pair of polytopes as below, they have the same number of facets (seven each) and they are combinatorially isomorphic.
Although the faces have different sizes when you compare the two, is it trivial that the facets of the first one are in one-to-one correspondence with the facets of the second polytope? If so, how does one find the map which creates the second polytope from the first one, is it some type of re-scaling of coordinates?
If neither edge sizes nor face parallelness are kept, the only rescue is the abstract polytopal incidence structure, either being outlayed as an incidence matrix (running on all the various elements, i.e. all vertices, edges, faces; cf. https://bendwavy.org/klitzing/explain/incmat.htm), or, equivalently, the abstract polytopal Hasse diagram, cf. https://en.wikipedia.org/wiki/Hasse_diagram). For, if these would coincide between the given 2 polytopes, then they definitely are equivalent as abstract polytopes too.
--- rk