On Tue, Jul 16, 2013 at 4:42 PM, Wayne Bishop <email@example.com>wrote:
> At 06:55 AM 7/15/2013, kirby urner wrote: > > As to how to duplicate a tetrahedron with a ruler and compass in 3D if >> that's what's asked, it sounds doable, but not in a way I'd want to try >> typing out in words here. Sounds like a CAD challenge of some kind. >> > > Assuming the traditional straightedge (Ax 1: Two points determine a line) > as opposed to a "ruler", I am sure it is not, and fairly easily reducible > to the traditional solution (or NON-solution, if you prefer), the > irreducibility of p(x) = x^3 - 2 over the integers (rationals, if you > prefer). You know, the contrapositive,"If the edge of a regular > tetrahedron could be constructed with compass and straightedge that is the > edge of a regular tetrahedron of double the volume of the original, then?" > > Wayne > > >
Yes, my apologies, not a "ruler" per se. My understanding is that, in practice, a straight edge was presented, as often as not, by a taut string, something easily carried around coiled up, yet usable for drawing any arcs or circles, so it's your compass as well (given line segments of 1/n subtend angles in proportion). Any cube contains a tetrahedron 1/3rd its volume as a set of face diagonals (two tetrahedra comprise all 12 diagonals) so whatever construction works or doesn't (if you prefer) in creating a cube with an edge/compass would have its associated tetrahedron implied.