Sorry for the poor title, but this question is awkward.
Numberphile has a new video about the plastic number, and it demonstrates it with a set of calipers with 4 prongs.
Towards the end of the video (11:10) , the guy remarked that there is a similar 3-pronged caliper that generates the golden ratio, but that "there is no way to do something similar with a 5-pronged caliper."
... why? It's by far the most interesting statement made in the video and they just say nothing at all and leave it at that.
Why can't something similar be done with a 5 pronged caliper?
edit: Here's a visual proof for this instance. I should have taken the time to draw this out before my initial post, sorry.(end edit)
While I have not yet come up with a general proof, I did manage to find a 5-prong counterexample, so his claim is incorrect. My counterexample contains the following ratios:
first layer: x^5, x^6, x^7, x^3 /
second layer: x^3, x^4, x^2 /
third layer: x^2, x^1 /
fourth layer: x^0 /
where 1=x^2+x^3
I think it is safe to conjecture that it is not possible to create calipers with an arbitrarily-large number of prongs, and I suspect the upper bound is no higher than 6.