I thought there might be some other interesting things to optimize here. Given a sphere, what's the largest exocube* it can be dissected into using *any* number of pieces? We include infinitely many, so we should say: What is the supremum of the exocubes that can be so made? [Aside: I was momentarily startled to see my friend's computer's autocorrect had changed exocubes to excuses, altering the sense a just a bit.] Alas, I think the answer is that an arbitrarily large exocube could be built. (((Hmm, what if you went the other way, from a solid cube to an exosphere? Again it looks as if there is no upper bound to the size of the exosphere.))) * * * But it might be interesting to minimize (find the infimum of) the *total area* of *all cuts*. Given a sphere containing volume = V, clearly the exocube must have side >= V^(1/3), so the TAC = total area of all cuts must be >= 6*V^(2/3). But that can't be the infimum, can it? —Dan ————————————————————————————————————————————————————————————————————————————— * Definition: ----------- An *exocube* looks like a cube from the outside and consists of the closure of a cube after their interiors of at most countably many disjoint closed topological 3-balls have removed from it. Allan Wechsler wrote: ----- It's obviously possible to dissect a sphere into pieces that could be reassembled into something that looks like a cube from the outside. Inside, it would be hollow, possibly with a bunch of spare pieces rattling around inside. Feasibility sketch: build thin polyhedral planks from the inner part of the sphere, with 45-degree bevels where necessary to let them form the cube's edges and vertices. All the extra material could be chopped up into manageable chunks and hidden in the interior. If there's too much extra material, redesign with thinner planks to make the outer cube bigger. Intuitively, it feels like we could get away with just a few dozen pieces or so: maybe 4 to 6 per face, and then an approximately equal number to pack inside ... but I don't know. Can anybody provide an explicit construction, or make lower-bound arguments about the number of pieces? -----