MINSKY%OZ.AI.MIT.EDU@XX.LCS.MIT.EDU (10/31/86)
Gary Allen's buckling-sphere argument seems convincing at first, but can one really prove that a shpere is indeed the best geometry? Gary says, "The geometry that can best withstand compression is a sphere." However, I suspect that this is only "locally" true for homgenous materials, and the theorem does not apply to inhomogeneous - let alone, fractile - materials. For example, if you made a pressure-bearing container of solid polystyrene, I don't doubt that the best you could do would be to form it into a sphere. But wouldn't it be vastly more resistant to buckling if you made it into a much thicker spherical shell composed of styrofoam?
MINSKY%OZ.AI.MIT.EDU@XX.LCS.MIT.EDU (11/01/86)
(Addition to previous note.) I did not mean to suggest that Gary Allen's argument is incorrect, but only to wonder whether the classical derivation considers every possibility. Presumably, it uses a variational method that finds the extremum for variations in the mass distribution of deviations from a spherical form and (correctly) finds the shell to be optimal. Very likely, this is correct, but I wonder if the variational method extends to distributed variations in density/porosity. Eric Drexler showed me a draft calculation that showed that it is feasible to make a lighter-than-air object with perfect carbon fibres. However, this is entirely in accord with Gary's argument since (1) those ideal fibres are indeed a couple of orders of magnitude better than steel and (2) Drexler's calculation did not suggest any large margin beyond that. Drexler's construction involved a dense tetrahedral lattice that supports a spherical, airtight shell. As I recall, Drexler was not maintaining that the hollow lattice was superior to the uniform shell in regard to preventing buckling. What he did argue was that the lattice could be constructed with so fine a grain that the fibres would be smaller than wavelengths of light. The resulting floating object, then, might also be invisible! Very cute, if true, but I don't know enough wave theory to know whether it would end up with a substantial refractive index, in any case.