Of the Platonic solids, only the cube can fill a 3D space. People often claim that the tetrahedron is also space-filling (including Aristotle, in his work On the Heavens) but, despite the fact that this, er, ‘fact’, is repeated again and again, it isn’t, in fact, a fact: it isn’t true. If feels like it should work but, if you stack tetrahedrons, there will always be some small gaps between them.
