Rich asks these tetrahedron questions: << (a) Given the areas of the four faces, what's the maximum volume? (b) What extra information is needed (beyond the face areas) to determine the volume? One or two edges? One of the lines connecting centers of opposite edges? Recall that altitude + opposite face determines the volume. Also that three face areas of a right tetrahedron determine the volume. So maybe we should be thinking about angles, or solid angles.

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Some other worthy questions are: (c) What are the possible 4 face areas of a tetrahedron? (To avoid extraneous info, one could assume the total surface area is constant, and the face areas given in non-decreasing order -- leading to the quotient of a regular 3-simplex by S_4 -- before finding which subset of this answers the question.) (d) What are the possible 4 solid angles at the corners of a tetrahedron? (Taking the exterior solid angle may be more convenient, since these add up to 4pi.) (e) What are the possible lengths of the 6 edges of a tetrahedron? (This would amount to labeling the 1-skeleton of a tetrahedron with lengths that can be realized. Rich once showed that satisfying the triangle inequalities for each face is not sufficient, as may have been mentioned here in the past.) --Dan _____________________________________________________________________ "It don't mean a thing if it ain't got that certain je ne sais quoi." --Peter Schickele