Neighbourly Polyhedra

k4In the plane, it is relatively easy to find four convex polygons which pairwise share an edge and are otherwise disjoint. Can you find five polygons in the plane with these properties? How about polyhedra? How many polyhedra can you find such that any pair of them share a face and are otherwise disjoint?
For me, this is an example where my three-dimensional imagination fails utterly. If you have never thought about this before, I suggest to try finding as many such polyhedra as possible (why not start with seven?) before reading on.

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