Subject: Re: [math-fun] left vs. right From: "Stephen B. Gray" <stevebg@roadrunner.com> Here's an exercise in 3D visualization. Given a point P, is it possible to construct FIVE rays coming out from P such that every ray makes an obtuse angle (>90 degrees) with every other one? (That's 10 angles that must be obtuse.) Explain your answer. Steve Gray ________________________________ It can't be done. Given one ray, v1, the other four must lie on the opposite side of the plane through P normal to v1. With v2 also chosen, the remaining three are confined to the inside of a dihedral with acute angle. With v3 also chosen, the remaining two are confined to the inside of a trihedral whose three dihedrals are acute. Let u1, u2, u3 be rays along the edges of the dihedral. Since v4 and v5 must be positive linear sums of u1, u2, u3, it suffices to prove that the dot products ui.uj are positive, for then v4.v5>0, and their angle cannot be acute. If we consider the spherical triangle intercepted by the trihedral, this becomes: "If the angles of a spherical triangle are acute, then their sides are acute." Note that the sides are measured by the angle they subtend at the center. I've pretty much forgotten my spherical trigonometry, so I will quote from ( http://mathworld.wolfram.com/SphericalTrigonometry.html ), eqs. (18)-(20). cos A = - cos B cos C + sin B sin C cos a, etc. by permutation. The angles A, B, C are acute, so their sines and cosines are positive. Whence cos a is positive, and so side a is acute, and similarly for the other sides, completing the proof of the assertion. -- Gene