Oops. I think the original question was: if each face of the tet has constant area, can the tet have different volumes. (Without rearranging the faces.) I somehow changed the question to: if the sum of the face areas is constant, can the volume vary. The answer to the second question is yes, but I don't know the answer to the original problem except as Gene gave it. Steve Gray -----Original Message----- From: math-fun-bounces@mailman.xmission.com [mailto:math-fun-bounces@mailman.xmission.com] On Behalf Of rwg@sdf.lonestar.org Sent: Wednesday, October 22, 2008 3:28 AM To: math-fun Subject: Re: [math-fun] tetrahedron volume
----- Original Message ----
From: "rwg@sdf.lonestar.org" <rwg@sdf.lonestar.org> To: math-fun <math-fun@mailman.xmission.com> Cc: math-fun <math-fun@mailman.xmission.com> Sent: Monday, October 20, 2008 3:37:08 AM Subject: [math-fun] tetraroller volume
True or False Quickie: The volume of a tetrahedron is determined by the areas of its faces. _______________________________________________
False. There exists a face area preserving continuous deformation that alters the volume. Start with a regular tetrahedron. Perform a dilatation along the mutual perpendicular of two opposite edges, while simultaneously performing a dilatation in the transverse plane so as to preserve the face areas. This is possible because all four faces are inclined with respect to that mutual perpendicular by the same angle. The volume is not preserved, in particular the volume goes to zero as the tetrahedron is squeezed flat. Gene
Yes. Or drawn to a long needle. There are formulas germane to other bits of this thread in the Mathworld tetrahedron article. It was surprisingly easy to write vol_polyhedron(faces) in terms of vol_pyramidpts(apex,face), in turn in terms of vol_tetrahedron(pts). It even flips your faces for you. For the Szilassi holeyheptahedron [[[-90,50,40],[75,75,-60],[-40,100,-160],[-240,0,240],[240,0,240],[140,50,40 ]], [[-140,0,40],[-140,-50,40],[-240,0,240],[-40,100,-160],[0,252,-240],[0,-252, -240]],[[-75,-75,-60],[75,75,-60],[-90,50,40],[-140,0,40],[0,-252,-240],[40, -100,-160]],[[75,75,-60],[-75,-75,-60],[90,-50,40],[140,0,40],[0,252,-240], [-40,100,-160]],[[140,0,40],[140,50,40],[240,0,240],[40,-100,-160],[0,-252, -240],[0,252,-240]],[[90,-50,40],[-75,-75,-60],[40,-100,-160],[240,0,240], [-240,0,240],[-140,-50,40]],[[140,50,40],[140,0,40],[90,-50,40],[-140,-50,40 ], [-140,0,40],[-90,50,40]]] it claims vol_polyhedron(%) = 22124800/3, if anyone cares to check. --rwg INCONSISTENT NONSCIENTIST PS, Mma 6.0 displays and tumbles the Szilassi perfectly! But I'm surprised it seems to have no AreaPolygon[pts_List], let alone VolumePolyhedron[faces_List]. _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun