A convex polytope (or other shape, but I'll concentrate on convex polytopes) P can be "Ruperted" if it is possible to drill a hole in it, then pass a copy of P thru the hole. Actually I'll only consider "line Rupertability" where the motion has to be along a line by pure translation; no twisting and bending motions allowed. SIDE QUESTION: Can anybody devise some examples of shapes P which can be Ruperted, but only in a non-line manner? ANSWER: I suspect such shapes do exist! Consider a "thickened helix", e.g wind some wire, of thickness 1 cm, round a helix. I think this shape can be Ruperted in a limit sense where the "drilled hole" is actually the entire shape shaved by epsilon, with the motion necessarily being a helical slido-rotation -- but any straight-drilling attempt disconnects? Also, instead of a helix, another possibility is a "thickened circular arc." You may object that construction was silly and degenerate. However, it may be possible to avoid that objection either by going to higher dimension or in 3D by making the "wire" have noncircular cross section. The point I am trying to drive at, is that for some shapes, it is very natiural to try to worm them thru twisting and/or curving drilled holes, and you pay a major penty if you try to drive them stright thru holes via pure translation. Specific 3D try: consider a thickened unit-radius semicircular arc in the XY plane, cut out of a piece of sheet metal so it has 3D thickness. Choose the sheet-thickness and the radial thickness right, that's 2 parameters. Well, I think this shape is (if parameters chosen right) just barely Rupertable, but NOT by any pure-translation motion, it needs to be a nonlinear motion. Crude Sketch: https://dl.dropboxusercontent.com/u/3507527/NonlinRupert.png Note, this object is nonconvex. I have no idea whether convex P could require nonlinear Rupertizing. RETURN TO MAIN TRACK (Linear Rupertability only): Greg Huber got in touch with me & Gosper, and anyhow in emails with him I outlined proofs the n-octahedron and n-simplex both can be linearly Ruperted for each n>=3; and the n-cube also is Rupertable as was already known. He sent a proof, which was much like a post by me but done better 23 years earlier that indeed the n-cube is Rupertable with shrink factor 1+const/9^n. The n-sphere of course is NOT Rupertable, and the status of the sporadic regular polytopes (icosahedron, dodecahedron, 600-cell, 120-cell, 24-cell) is presently unknown but could be solved by computer by the method I will now describe. Here is how to decide if convex polytope P is Rupertable. Choose a direction d. Project the vertices of P along d onto a plane perp to d, call that operation Proj[d](P). Also choose a rotation (or orthogonal, if we allow mirror reflect...) matrix Q. P is Rupertable in direction d using Q if and only if the vertex set of Proj[d]( Q P ) is translatable to lie inside ConvexHull( Proj[d](P) ) -- which question is a linear programming problem once d and Q are given. Now perform that test for lots of d and Q, covering the space of rotation matrices and sphere of directions with a dense set and trying them all until you find a successful test. Then you have proven it Rupertable. This will always eventually succeed (with probability 1 using random points...) in proving Rupertability if it is Rupertable, because the spaces to be searched are compact manifolds and the success set is algebraic and hence has nonzero measure, so there is a positive probability of success each try. Now if P is not Rupertable, then this kind of crude approach, rather less obviously, can eventually prove that too, in finite time. Because: we can produce, a priori, some rather weak but valid degree and "height" bounds on the algebraic set. We then try as our set of trials, all algebraic numbers of low degree and height... or actually any dense-enough finite set of points (using known "separation" bounds for algebraic numbers of given degree and height arising from discriminants...) if they all fail, that meant there were no solutions. -- Warren D. Smith http://RangeVoting.org <-- add your endorsement (by clicking "endorse" as 1st step)