I typed Bill's hexad below into my program, which sure enough reported 3 small tetrahedra (out of the 30 possible) with absolute volumes 15.32733125 , 35.63804829 , 46.20211861 However, among the remainder were another 5 with volumes around 100, followed by a steady ramp up to the largest around 1000. Run-of-the mill integer hexads I've generated frequently show distinct volumes much closer together than these figures. So taken in the round, I have to say they're not terribly convincing. The search program has been souped up to generate integer hexads restricted to the region where all 20 possible faces are proper triangles. In short test runs it has so far turned up a handful of double proper tetrahedra (meaning two distinct shapes with the same volume and edge lengths), all in sets of threes with the same hexad; but not even a single planar hexad for edge-sum up to 288. It ain't looking good! Fred Lunnon On 11/27/06, Bill Thurston <wpt4@cornell.edu> wrote:
... Looking back at the message I sent, I see now that the text I copied and pasted from GSP was actually a TIFF graphic, and no doubt the math-fun-gremlins ate it up. I'll type it in: AB = 36.38 cm AC = 16.29 cm AD = 21.06 cm BC = 28.31 cm BD = 15.47 cm CD = 14.64 cm where C,.5(A+B),D and C,.5(A+D),B are (approximately) right angles.