So I was watching Numb3rs this evening with my roommate while doing a little proposal writing. Numb3rs is a ridiculous show where fairly complex methods are used to solve crime. The show itself is barely watchable — it’s like they just open up a math text and write down whatever looks most complex while using absolutely ridiculous metaphors to incorrectly explain mathematical concepts. It’s the CSI of math and physics. However, at the end of an episode they discussed the fact that it is impossible to bend a piece of spaghetti and break it into only two pieces. Feynman was known to investigate this phenomena but was actually unable to explain it. I took it upon myself to do my own experiments. That and a little theory are documented below.

So: angel hair pasta. It’s a cylindrically symmetric britle rod which bent beyond the limit of curvature does, in fact, break into N > 2 pieces. I have verified this by experiment. So, Numb3rs got something right. I had some Barilla fettuccine pasta lying around so I decided to give it a try. Fettuccine pasta is only planar symmetric (those who know their group theory will recognize this as a lower point group), and when bent along the plane of the pasta beyond the radius of curvature it as well breaks into N > 2 fragments. (When bent in the orthogonal direction it always breaks into exactly three pieces! And the third piece is always a small fragment only in that direction! Go ahead, give it a try. You’ll see. This is probably explainable by the fact that a defect in this direction causes a small fragment followed by a clean full break, although I have no specific theory applicable to this problem.)

Anyway, back to the point. I was sufficiently intrigued to look up any theory on the subject. Feynman died ( 1988 ) before any explanation of this phenomena was available, as the Kirchhoff equations had not yet been developed ( 1993 ). For small planar oscillations these equations take the form:

where the commas in the indices denote partial derivatives. The solutions to these equations indicate that when a small britle rod is subjected to a bend right before the radius of curvature and released, the subsequent curvature in sections of the rod exceed the initial curvature and may induce fragmentation! This seemed pretty counter-intuitive, so I gave it a try with some fettuccine. Indeed, this is the case. These results are summarized in the figure below.

Here, is the initial radius and is the current curvature. Clearly we can see that even though we never physically bent the rod past the limit of curvature, the subsequent curvature after release exceeds that of the initial.

This fully explains our result. Bent past the limit of curvature one break is immediately followed or occurs with an augmentation of the curvature at points along the rod, causing further fragmentation.

If you would like to know more about all of this you can get some info and some nice Quicktime video here. The article that the math and figures presented here came from is here. I hope this random Numb3rs-induced digression was entertaining and informative for you all. Cheers.