By "opposite" I think Julian means reflected in a horizontal plane. --rwg ---------- Forwarded message ---------- From: Julian Ziegler Hunts <julianj.zh@gmail.com> Date: Sun, Sep 25, 2016 at 7:55 PM Subject: Re: [math-fun] Bar bet To: Bill Gosper <billgosper@gmail.com> But don't we need to prove that it _is_ tetrahedral, i.e. the tetrahedron
and octahedron dihedrals are supplementary?
Easier to just intersect two tetrahedra and note that this gives an octahedron with all the symmetries of the regular octahedron. How do we know that the intersection is an octahedron? What we're actually doing is taking a regular tetrahedron and cutting off the vertices to get an octahedron, then noting that the planes we're using to cut them off have the same symmetries hence also form a regular tetrahedron, which by inspection is opposite to the original (so that you can swap them). Julian