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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].