Can you make an icosahedron out of tetrahedra?
No - at least not assuming you mean a solid regular icosahedron and
regular tetrahedra. [...]
No, I'm not sure I meant a solid icosahedron. I
was only thinking as
far as a tetrahedron has equilateral triangles as faces, and so does
an icosahedron.
Then yes, there's no reason you couldn't take 20 tetrahedra and
position them so as to result in something that's basically a
tetrahedron constructed, inward-pointing, on each face of an
icosahedron. (It might not be all that easy to realize physically, but
that's a separate issue.)
I'm not sure it would be worth doing, though; the tetrahedra would
touch one another only along the edges of the icosahedron, limiting the
communication possibilities.
You could, of course, use non-regular tetrahedra formed such that each
one can be described as three vertices plus the centre of an
icosahedron; then they'd fit snugly (but not be good for much else).
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