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Background: The basic question as given in 'Research Problems in Discrete Geometry' By Moser, Brass and Pach (page 98) is: What is the max number of congruent infinite circular cylinders that can be arranged in 3d space so that every pair is touching? Several partial results are known including a proof by Bezdek that this number cannot exceed 24.

Questions:

  • What if we relax this question to infinite circular cylinders with not necessarily equal radii? What happens if we look at congruent and infinitely long prisms all with same cross sections (triangular, square etc..)?
  • If each infinitely long object only needs to be of uniform cross section but could have mutually differently shaped cross sections what can one say?
  • Can one have an arbitrarily large number of convex and mutually congruent solid objects that can all touch one another?

(Note: last question above is known to have "yes" answer as given in the comments below).

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    $\begingroup$ I have voted to close because this needs more focus. There are at least five questions here, mostly open-ended, with no clarity on what would make for an accepted answer. $\endgroup$
    – user44143
    Mar 12, 2021 at 15:38
  • $\begingroup$ To get a sense of how difficult some of these questions might be, consider the analogous question, How many unit cylinders can touch a unit ball?, and @MoritzFirsching's impressive analysis. $\endgroup$
    – Joseph O'Rourke
    Mar 12, 2021 at 17:14
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    $\begingroup$ @MattF. I think it's fine to ask multiple questions as long as they're all closely related -a positive or negative answer to one would imply a positive or negative answer to some of the others and it would be reasonable for the question to count as resolved. But it wouldn't hurt to list one of the questions as most important - e.g. "Do there exist for all natural numbers $n$ connected sets $S_1,\dots, S_n$ in $\mathbb R^2$ so that we can rotate and translate $S_1 \times \mathbb R, \dots, S_n \times R$ so that they all touch?" $\endgroup$
    – Will Sawin
    Mar 12, 2021 at 23:47
  • $\begingroup$ I've seen the answer 7 for semi-infinite cylinders (actually matchsticks) in at least one puzzle book but haven't ever sought out any known upper bounds. $\endgroup$
    – Ben Barber
    Mar 13, 2021 at 14:33

2 Answers 2

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"an arbitrarily large number of objects that can all touch one another"

I think one must be careful to define "object" and "touch." Otherwise, all these $n$ line segments (objects) intersect (touch) at the origin:

     enter image description here

But this example violates the spirit of the OP's questions.

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    $\begingroup$ An example that is less against the spirt, but may still not be entirely according to the spirt, consists of $n$ infinitely long triangular prism, where the triangle is acute of angles at most $2\pi/n$, joined together around a line, so their cross-sections look like a pie sliced into wedges. It's not immediately obvious to me how to give a precise formulation avoiding this example, if one wants. $\endgroup$
    – Will Sawin
    Mar 13, 2021 at 2:12
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    $\begingroup$ This would be better as a comment than as an answer, even if we have to visualize the line segments through the origin in our heads. $\endgroup$
    – user44143
    Mar 13, 2021 at 2:12
  • $\begingroup$ Thanks for pointing out an imperfection in the way the last question was phrased. I had meant convex objects with finite dimensions, not straight lines. $\endgroup$
    – Nandakumar R
    Mar 13, 2021 at 18:14
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For the last question, it’s easy to get an arbitrarily large number of objects in $\mathbb{R}^3$ that all touch each other. Imagine a stack of $n$ unit cubes. Next to the stack, you can place $n$ $1/n \times 1 \times n$ boxes such that each one touches all the cubes. Adjoin each box to one cube, and you’re done.

[Note: This answer applied to a previous version of the question.]

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    $\begingroup$ You can even achieve this with convex solids, as noted to me by Douglas Zare in this comment: mathoverflow.net/questions/97933/touching-tetrahedra-graphs/… $\endgroup$
    – Yoav Kallus
    Mar 13, 2021 at 2:44
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    $\begingroup$ @YoavKallus, that comment refers to the paper here: jeffe.cs.illinois.edu/pubs/pdf/crum.pdf $\endgroup$
    – user44143
    Mar 13, 2021 at 14:28
  • $\begingroup$ Thanks All. I was looking at the possibility of infinitely many *convex solids * that touch each other. Thanks for the pointer. Improved the question wording. $\endgroup$
    – Nandakumar R
    Mar 13, 2021 at 18:23
  • $\begingroup$ @NandakumarR: And now the answer is Yes, established in the Erickson-Kim paper cited in the comments. $\endgroup$
    – Joseph O'Rourke
    Mar 13, 2021 at 18:46
  • $\begingroup$ Sure. Shall note this in the question itself. Thanks. $\endgroup$
    – Nandakumar R
    Mar 13, 2021 at 19:09

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