Only a few polyhedra are flexible. It has been proved that flexible polyhedra must have constant volume and this is indeed true of all these polyhedra. In fact, they all have a volume of exactly half of space. Also, they are all self (or reverse) complementary.
To see these polyhedra flex, drag the angle slider. Note that some of these polyhedra at certain angles will gain greater symmetry and in these forms will be listed elsewhere as a different polyhedron.
These flexible polyhedra remain uniform as they flex. There are also other polyhedra that are also flexible, but the difference is that they do not remain isogonal.