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Starting in the center of a sphere of radius 1, draw a path with the shortest possible length that intersects every plane that is tangent to the sphere.

This question appeared as a generalization of the recently considered problem of the lost ant

Starting in the center of a circle of radius 40 ft, draw a path with the shortest possible length that intersects every line that is tangent to the circle.

to the third dimension.

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    $\begingroup$ This question looks much harder even than that one, and that one was only answered by reference to a paper in French (as I recall)! $\endgroup$ Commented Dec 23, 2013 at 18:59
  • $\begingroup$ @dfeuer It's harder to get an exact solution of this problem but I hope it would be easy enough to get some approximate solutions. $\endgroup$ Commented Dec 23, 2013 at 19:08
  • $\begingroup$ My bet is that the solution will start with a straight segment exiting the sphere (of course) and then spiral around toward somewhere near the opposite pole, probably doing something a bit funky at each end. I have no clue what sort of spiral that will be, exactly. Coming up with a way to calculate whether a given path offers enough coverage seems tricky (when considering computational approaches). $\endgroup$ Commented Dec 23, 2013 at 19:42
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    $\begingroup$ Well geez. Why not an $n$-sphere, while we're at it? $\endgroup$ Commented Dec 24, 2013 at 1:37
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    $\begingroup$ @AlexanderGruber - sounds like you've got another thesis topic :) $\endgroup$ Commented Dec 24, 2013 at 9:14

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This isn't an answer to your question, but doesn't a sphere have infinitely many tangential planes? This is true about a circle and tangent lines, but how to you walk the surface area of a sphere? This will take a path that is infinitely long, unless you assign a width for the ant's path.

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  • $\begingroup$ That such paths exist isn't an problem. Whether or not they have a length is another matter. $\endgroup$ Commented Apr 19, 2014 at 16:48

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