"AI Overview" in Chrome says a polyhedron is:
"a three-dimensional geometric shape with flat polygonal faces, straight edges, and sharp corners or vertices"
which is not a bad definition in my view. Some of these polyhedra are familiar to us; for example here is a back-of-an-envelope sketch of some with 4, 5 and 6 faces.
Note that we are not concerned here with the particular angles or side lengths, just with the topology: how many faces and vertices there are, how many sides each face has, and what faces fit with what other faces around each vertex.
The solid with 4 faces - a tetrahedron aka triangular pyramid - is the only possibility for 4 faces, and for 5 faces there are exactly two polyhedra - the pyramid with a quadrilateral base and the triangular prism.
The two hexahedra shown are not the only ones however. There are 7 altogether; can you find some or all of the remaining ones? Solution in the Comments.
As the number of faces grows the number of possible polyhedra climbs exponentially [1].
| 4 | 1 |
| 5 | 2 |
| 6 | 7 |
| 7 | 34 |
| 8 | 257 |
| 9 | 2606 |
| 10 | 32300 |
| 11 | 4.4E+05 |
| 12 | 6.4E+06 |
| 13 | 9.6E+07 |
| 14 | 1.5E+09 |
| 15 | 2.4E+10 |
| 16 | 3.9E+11 |
| 17 | 6.4E+12 |
| 18 | 1.1E+14 |
Numbers up 10 faces are exact, but those from 11 on are only estimated.
Amazingly Steven Dutch has enumerated and classified all 2606 enneahedra, finishing on 2 November 2016 [2].
Here's a quick question to finish - what is a regular hexahedron better known as?
References
[1] https://oeis.org/search?q=A000944&language=english&go=Search