3
$\begingroup$

Is it possible to trap a sphere in the center of a trefoil knot? It seems like with three points of contact it should be possible. Admittedly not big into math, but a craftsman who loves trapping marbles. Sincerely appreciate any thoughts or suggestions.

$\endgroup$
3
  • 1
    $\begingroup$ It takes four points to trap a sphere. $\endgroup$
    – TonyK
    Sep 9 '14 at 12:13
  • 1
    $\begingroup$ Also posted to MO, mathoverflow.net/questions/180440/…, without notice to either site. $\endgroup$ Sep 9 '14 at 12:43
  • $\begingroup$ Thank you Tony. Maybe if I give the inside lines a heavy curve to touch more surface area it could work... And yes, Gerry, I posted it to overflow first, and it was mentioned this might be a better resource for me. $\endgroup$
    – Ross
    Sep 9 '14 at 13:06
1
$\begingroup$

This depends on where you want to trap the sphere. Usually, a trefoil goes around the outside (major radius) of the torus twice and smaller (minor) three times. We sometimes call this a (3,2) torus knot. The best place to put a marble or sphere would be the center of the torus, but this only will have three points of contact, and as TonyK stated, we need 4 if we want to hold the sphere.

But, if we use the trefoil where we switch the meridian and longitude, we get three "strands" which flow around the torus, a (2,3) torus knot. See some useful pictures here. Then a few marbles could roll around inside the area where the torus would be, and I believe, be trapped.

If you want to let the trefoil be in some other configuration where it doesn't necessarily lie in a torus, I am sure you can get it to hold a sphere stationary. Hope this helped.

$\endgroup$
0
$\begingroup$

Here is an example suggested by Scott Carnahan in response to a different question, "Hanging a ball with string," which I include in case it might yield some intuition.


          ScottCarnahan

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.