Can two regular tetrahedrons make an octahedron?

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I'm interested in how tetrahedrons can fill space. First of all, do two tetrahedrons of the same size make an octahedron (assuming you could arrange the tetrahedrons in any way you want, without strict mathmatical reflection, rectification, dualing, etc.)

If they do, it's known that tetrahedrons and octahedrons fill space perfectly together (2:1 ratio)

but really that would just mean 4 tetrahedrons, arranged in different directions, if I'm thinking correctly, even though they have different "names", could it all be constructed to fill space with simply regular tetrahedrons arranged in different directions?

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2 Answers

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An octahedron has 8 faces, by definition. A tetrahedron has four, by definition. If you attach two tetrahedra together such that a face from one completely meets a face from the other, you have six faces left on the resulting solid, so you can't have an octahedron.

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You can make a stellated octahedron by converging two tetrahedrons.

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