A tetrahedron or triangular pyramid.
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Bipyramids are a class of polyhedra with more faces than vertices.
There are many such polyhedra. See the link below. The classification and naming of polyhedra is not wholly satisfactory so it is not possible to simply provide a list of names.
The simplest stellated polyhedral shape is the stellated tetrahedron with 12 faces (3 * 4) Others have more faces.
There are only 5 regular polyhedra: those with 4, 6, 8, 12 and 20 faces. If you know of 7 polyhedra there may be a Fields Medal (the Nobel prize for mathematicians) for you!
Regular polyhedra have identical faces.