Timeline for How many unit cylinders can touch a unit ball?
Current License: CC BY-SA 3.0
7 events
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| Mar 31, 2014 at 18:48 | comment | added | Wlodek Kuperberg | @BenoîtKloeckner You're right: the points of contact would remain antipodal. But none of the cylinders would remain parallel. So, even if Step 1 were correct, Step 2 could still fail. | |
| Mar 31, 2014 at 18:26 | comment | added | Gerhard Paseman | In particular, consider moving one 2-tile continuously against a fixed 2-tile so that the corresponding cylinders stay tangent. I think you will that the only motions for two cylinders that might allow a radical change in configuration lead to a large separation of contact points and thus that one or more cylinders will be squeezed away from the central sphere. Gerhard "Ask Me About System Design" Paseman, 2014.03.31 | |
| Mar 31, 2014 at 18:17 | comment | added | Benoît Kloeckner | @WlodekKuperberg: if the two triples of cylinders move symmetrically, then this movement is satisfying step 1. This does not imply that it cannot happen, but that the statement "step 1 => lockedness" would rule out such a movement. | |
| Mar 31, 2014 at 18:16 | comment | added | Gerhard Paseman | Consider making six copies of a spherical "2-tile". The bottom layer of the tile is that section of a sphere between the center and 'the closest' cylinder, and the second layer represents the section of the sphere between the center and any point of the same cylinder. Note the first layers remain disjoint, and the second layers can slide over the first layers of a different tile. Now try moving these tiles around on a basketball. Gerhard "Solving Problems Using Analog Computers" Paseman, 2014.03.31 | |
| Mar 31, 2014 at 16:10 | comment | added | Wlodek Kuperberg | There is an "invisible" cube circumscribed about the ball in Figure 3; look at the three cylinders "adjacent" to the cube's front-upper-right vertex. Sometimes I think that the configuration of the three cylinders can be "twisted" continuously, so that their points of tangency move closer to the said vertex, allowing for the other three cylinders to twist symmetrically to the ball's center, while all six cylinders remain non-overlapping, but I have no patience for the necessary computations... Maybe someone would like to try a computer experiment...? | |
| Mar 31, 2014 at 15:56 | comment | added | Wlodek Kuperberg | Step 1 is crucial, and not obvious at all. I have mixed feelings about it: sometimes I feel it must be that way; sometimes I don't. | |
| Mar 31, 2014 at 15:17 | history | answered | Benoît Kloeckner | CC BY-SA 3.0 |