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Many years ago I picked up a little book by J. E. Littlewood and was baffled by part of a question he posed:

"Is it possible in 3-space for seven infinite circular cylinders of unit radius each to touch all the others? Seven is the number suggested by counting constants."

It is the bit in italics which baffled me then (and still does). Can anyone explain how he gets 7 by "counting constants"?

P.S. For completeness, the book is "Some problems in Real and Complex Analysis" (1968)

Old John
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    I do not know about this problem but there is a related problem for spheres posed by Donald Coxeter: see http://www.popmath.org.uk/sculpture/pages/5firm.html and the linked explanation. The answer for spheres is 5. I do not know if he ever considered cylinders. – Ronnie Brown Oct 31 '12 at 17:13
  • Many thanks. I definitely like that sculpture! – Old John Oct 31 '12 at 17:19
  • Should we think of the cylinders as being solid (like the spheres in the sculpture), or are they allowed to intersect? The terms "touch" implies tangency to me, but I thought I'd check. – Fly by Night Oct 31 '12 at 17:28
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    He doesn't make it clear in the book, but I would assume he means solid and that he is talking about tangency. – Old John Oct 31 '12 at 17:35
  • Yeah, otherwise you could just have them all intersect easily. – Christopher A. Wong Oct 31 '12 at 17:41
  • Put 6 cylinders around a central cylinder – i. m. soloveichik Oct 31 '12 at 18:59
  • But how can we be sure that each of the six cylinders touches each of the other 5? – Old John Oct 31 '12 at 19:01
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    http://demonstrations.wolfram.com/SevenCylinders/ – Cocopuffs Oct 31 '12 at 19:10
  • Nice animation! However, those cylinders are not infinite (in both directions), which I assume Littlewood intended - could it be modified to work with doubly-infinite cylinders? The real question in my mind, though is how Littlewood arrived at such a conjecture "by counting constants". – Old John Oct 31 '12 at 19:15

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Here is my take: There are $4$ degrees of freedom in selecting the center line of each cylinder, for a total of $4n$ degrees of freedom. Subtract from this the $6$ degrees of freedom given by the Euclidean motions (rotations and translations in space), as applied to the total configuration – for a total of $4n-6$ degrees of freedom.

For two cylinders to touch, the minimal distance between points on their respective center lines must be $2$. This results in $\binom{n}{2}$ equations. To be able to satisfy all these equations, we must probably have $4n-6\ge\binom{n}{2}$, which holds for $n\le7$.

Harald Hanche-Olsen
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