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Timeline for Cylinders dividing $\mathbb{R}^{3}$

Current License: CC BY-SA 3.0

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Aug 6, 2017 at 13:09 history edited Joseph O'Rourke CC BY-SA 3.0
Broken link fixed.
Jun 5, 2017 at 10:27 history edited Joseph O'Rourke CC BY-SA 3.0
Image link broken; now fixed.
Sep 20, 2012 at 12:13 vote accept Victor
Sep 19, 2012 at 21:02 comment added Gerhard Paseman And if you are gracious enough to humor me, how many regions do you get from a slight longitudinal rotation to a 3x1x1 cylinder? Gerhard "I'm Thinking You Get Ten" Paseman, 2012.09.19
Sep 19, 2012 at 20:57 comment added Gerhard Paseman I hope you don't mind this method of asking, Joseph. VCF (in a comment to my answer) posed the possiblity of up to 8 regions with two congruent ellipsoids. For the drama of contention (and through lack of imagination) I say six. Do you know what the answer is? Gerhard "Thank You For Your Attention" Paseman, 2012.09.19
Sep 16, 2012 at 13:27 comment added Douglas Zare Oops, I meant $\Omega(n^3)$ not $\omega(n^3)$ above.
Sep 16, 2012 at 12:38 comment added Joseph O'Rourke @Douglas: You are absolutely right. I stand corrected---Thanks!
Sep 16, 2012 at 12:35 comment added Victor @Joseph: It would be nice to have the kind of closed formula I am looking for. But the asymtotic complexity point of view makes me think that perhaps it isn´t necessary to have one.
Sep 16, 2012 at 2:57 comment added Douglas Zare In the first quote, the combinatorial complexity is $v+e+f$ on the boundary of the union of solid cylinders, not the number of regions formed by their surfaces. There can be $\omega(n^3)$ regions from infinite hollow cylinders, or any smooth surface, since there can be that many bounded regions from planes (as in the coins configuration).
Sep 16, 2012 at 2:40 comment added Gerhard Paseman I still have concerns, but they may be illfounded. It looks like helical cylinders will not be defined by finitely many algebraic surface patches. Thank you for expanding on this, and do not be distressed by my reservations and contrary position. Gerhard "Gaining Some Expertise Each Day" Paseman, 2012.09.15
Sep 16, 2012 at 0:48 history edited Joseph O'Rourke CC BY-SA 3.0
added 284 characters in body
Sep 16, 2012 at 0:42 history answered Joseph O'Rourke CC BY-SA 3.0