What is the smallest diameter ring a non-convex polyhedron can pass through in 3-space? - MathOverflow most recent 30 from http://mathoverflow.net 2013-06-19T21:20:38Z http://mathoverflow.net/feeds/question/82312 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/82312/what-is-the-smallest-diameter-ring-a-non-convex-polyhedron-can-pass-through-in-3 What is the smallest diameter ring a non-convex polyhedron can pass through in 3-space? UltraBlue06 2011-11-30T21:32:47Z 2011-12-01T02:40:29Z <p>The question is mostly in the title. </p> <p>Imagine I have some non-convex polyhedron \$P\$, and I would like to find the smallest diameter ring that it can pass through in 3-space, undergoing any necessary rotations as it does so. Is there an efficient way to calculate \$D_{ring}\$? Pressing my luck, can I find the set of rotations for \$P\$ as it passes through the ring? </p> http://mathoverflow.net/questions/82312/what-is-the-smallest-diameter-ring-a-non-convex-polyhedron-can-pass-through-in-3/82315#82315 Answer by Igor Rivin for What is the smallest diameter ring a non-convex polyhedron can pass through in 3-space? Igor Rivin 2011-11-30T21:48:53Z 2011-11-30T21:48:53Z <p>This is the "piano movers problem", also known as the motion planning problem, which has an enormous literature. Check out <a href="http://en.wikipedia.org/wiki/Motion_planning" rel="nofollow">http://en.wikipedia.org/wiki/Motion_planning</a></p> http://mathoverflow.net/questions/82312/what-is-the-smallest-diameter-ring-a-non-convex-polyhedron-can-pass-through-in-3/82343#82343 Answer by Joseph O'Rourke for What is the smallest diameter ring a non-convex polyhedron can pass through in 3-space? Joseph O'Rourke 2011-12-01T02:40:29Z 2011-12-01T02:40:29Z <p>Just a side remark on <em>convex</em> polyhedra: Each of the regular polyhedra except the cube can pass through a circle of radius smaller than the smallest-radius cylinder circumscribing the polyhedron. This is proved in Tudor Zamfirescu's delightful paper, "Convex polytopes passing through circles" (<a href="http://fmi.unibuc.ro/ro/teme_cercetare/cex06_gta/RaportareIV/PolyThroughCircles.pdf" rel="nofollow">PDF link</a>). There is quite a nice (non-algorithmic) literature on this problem.</p>