3
$\begingroup$

been stuck on solving/proving the following puzzle:

You need to make a hole in the wall, so that a 1 meter line can pass it through the hole at all angels, find a shape with minimum surface area that would satisfy the above conditions ?

$\endgroup$
  • $\begingroup$ A cone maybe, not sure $\endgroup$ – Kushal Bhuyan Nov 11 '15 at 7:20
  • $\begingroup$ Isn't this an unsolved problem? $\endgroup$ – Christopher Carl Heckman Nov 11 '15 at 7:33
  • 5
    $\begingroup$ Isn't this the Kakeya needle problem then? en.wikipedia.org/wiki/Kakeya_set $\endgroup$ – MarsOneRover Nov 11 '15 at 7:48
  • 1
    $\begingroup$ Doesn't the aforementioned Wikipedia article show that the minimum area (measure) is zero? $\endgroup$ – copper.hat Nov 11 '15 at 7:56
  • 1
    $\begingroup$ @copper.hat yes. $\endgroup$ – MarsOneRover Nov 11 '15 at 7:59
2
$\begingroup$

Take an equilateral triangle with sides of length $1$, and then use each vertex as the center of a circular arc passing through the other two vertices:

enter image description here

This is at least a contender, with area $3\cdot\frac{\pi}{6}-2\cdot\frac{\sqrt3}{4}$, less than a circle of diameter $1$ or a quarter-circle of radius $1$, which are other convex shapes meeting the description. This shape may be the winner if you require a convex shape, but I have no ideas for proving it.

$\endgroup$

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.