3
$\begingroup$

Let $T$ be a unit edge-length equilateral triangle composed of three cylinders each of (small) radius $r>0$. (By "small" I mean approximately $< 0.1$.) Think of $T$ as a physical, rigid triangle, perhaps made from polished steel.

Consider a string of linked such triangles, $T_1, T_2, T_3, \ldots, T_n$.

Q. When one tugs on a corner of $T_1$ and a corner of $T_n$, stretching the string in opposite directions, what is the preferred orientation of the triangles?

Below I depict a possible configuration, with every other triangle rotated $\pi/2$ with respect to its neighbors. But I have no confidence that this would be the taut chain configuration.

StringTriangles600
If anyone has an appropriate physical analog and could try this experimentally, I would be interested to know the empirical result. Thanks!

Improve this question
$\endgroup$
4
  • $\begingroup$ My experience suggests that one does not get a linear configuration as pictured, but has several kinks. Perhaps one gets 175 degrees between adjacent triangles instead of the pictured 180. $\endgroup$
    – The Masked Avenger
    Commented Oct 7, 2014 at 14:59
  • 2
    $\begingroup$ As for real life examples, you might search for necklace or bracelet images on the web. $\endgroup$
    – The Masked Avenger
    Commented Oct 7, 2014 at 15:04
  • 2
    $\begingroup$ Perhaps consider approximating the "no crossing" condition with a smooth repulsive potential energy. Also approximate the tautness with an energy which grows with the total length. See what if anything can be said when limits of infinite stiffness are taken. Tge solution should minimize these energies $\endgroup$
    – hoj201
    Commented Oct 7, 2014 at 16:28
  • $\begingroup$ It might be relevant to specify where exactly the green arrows at the end are attached, since the ends are not points, but do have thickness. Steel has weight, so then the catenary shape gets involved (this might be irrelevant for taut chains). Finally, I have no reason to believe that what looks like two dashed parallel straight lines (one of blue triangle sides, the other of red triangle sides) would indeed be straight (or if the pieces would be parallel), in particular if the chain has only two or only three triangles (perhaps start with this case). $\endgroup$
    – Mirko
    Commented Oct 31, 2014 at 16:48

0

Reset to default

You must log in to answer this question.