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Archive for February, 2012

Driving Me Nuts

Posted by Patrick on February 16, 2012

Yesterday, Peter (@polarisdotca) asked this question:

Why does tying knot in strip of paper form a regular pentagon? Why not 6, 7,…? Why regular? Anyone have intuitive explanation? #wcydwt [1]

Being a rock climber, I like knots; I DEPEND on knots!  Being a math and physics teacher, I like puzzles; I DEPEND on puzzles.  So naturally, this one peaked my interest.  Here’s what I’ve got so far:

The first step was to recreate the experiment, so I started by making a regular knot (actually called the “Overhand knot” [2]) with a strip of paper:

Then, I tried to flatten it as tightly as possible without breaking it:

It’s a little loose at the “exit points”, but we can easily imagine that the “ideal case” would indeed be a regular pentagon (regular because all sides are the same lengths; pentagon because it has five sides).  So now: why is that?

Intuitively, I think there can only be five sides because there are three folds and two exit points, for a total of five.  That’s how the knot is made, by folding the rope three times onto itself:

Here’s what it looks like when unfolded:

Three of the sides are from folding, and two of the sides are just the edge of the strip of paper, which correspond to the exit points.

Why does it have to be regular though?  Is it because that’s the most compact configuration?  Is this shape the solution to some optimization problem (like greatest ratio of SurfaceArea-to-Perimeter, which minimizes some energy function or something…)

My next question was: how would a Figure-Eight knot [3] behave?  I was not only interested in this knot because I probably use it more often than the overhand knot, but because my trick to make it is to start it like an overhand knot, then finish it an extra half turn later (ie. that would add an extra fold in the strip of paper!)  Could this lead to a 6-sided figure?
Here it is loose:

And flattened:

Yeap: four folds and two exit points.  Here’s the weird thing though: one of those exit point is not even “connected” to the other sides:

Why is that?!  Also if I could make it perfectly, would it also be a regular polygon? or is it intrinsically elongated?  Thanks Peter!  This puzzle is driving me nuts!

Links:

  1. Peter Newbury’s Tweet:
    <https://twitter.com/#!/polarisdotca/status/169583691893444608>
  2. Animated Knots, Overhand Knot,
    <http://www.animatedknots.com/overhand/index.php>
  3. Animated Knots, Figure 8 Bend,
    <http://www.animatedknots.com/fig8join/index.php>

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