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12 Mar

Why cats land on their feet - a problem of 80 years standing!

I just made an amazing discovery while reading a biography of Vito Volterra.

Naturally, the book mentions many other mathematicians and describes their work. What arose my interest was a mention of a paper by Giuseppe Peano "Il principio delle aree e la storia di un gatto" (The principle of areas and the story of a cat.) The paper appeared in Rivista di matematica.

The article referred to the discussion that had taken place at the Académie des sciences in Paris about why a cat, no matter how it falls, always lands on its feet. The question has arisen after a series of photographs had provided documentation of the dynamics of the fall.

It's not clear from the book whether, and if so then how, the Académie des sciences in Paris has settled the question such that Peano's solution might well be original. I do not know if you ever tried to find an explanation to the notorious ability of the feline species to land on their paws. Last year I had a chance to ponder that question while enjoying Mark Levi's book which bore the question as the title:

Curiously, Peano's explanation was wrong and Mark Levi gave a reason why it was so. Peano's brought an engagingly plausible argument:

this animal left to its own devices, describes with its tail a circle in the plane perpendicular to the axis of its body. In consequence, by the principle of areas, the rest of its body must rotate in the direction opposite to that of the motion of its tail; and when it has rotated by the desired amount, it stops its tail and with that arrests at the same time the rotational motion, saving in that way both itself and the principle of the areas.

Levi blithely refutes this kind of argument:

Some people say that the cat does that by spinning the tail. On close inspection this turns out to be false. As an experimental fact, tailless cats are just as good as tailed ones in flipping over. Alternatively, a theoretical argument shows that to accomplish a 180° flip in a fraction of a second, the cat would have to spin its tail so fast that its tip would have to break the sound barrier, or to come close. This would create a sonic boom, or a loud whistling at the very least. And enormous centrifugal force would cause a part of the tail to tear off and become a deadly projectile, almost like a bullet. So the "tail" theory quickly flunks the sanity test.

The question appears to be of broad interest. There even are two wikipedia pages that shed light on the subject Cat righting reflex and Falling cat problem both of which answer the question in agreement with Levi's book.

Moreover, advances in high-speed photography allowed National Geographic to produce a film that leaves no doubt of the veracity of the later-day answer. Admittedly, the French Academy was at a great disadvantage trying to tackle the problem some 125 years ago.

So, what is the right answer? As seen in the movie, the cat first bends in the middle, then twists so that its halves spin in different directions (likely in conformity with Peano's principle of areas, but also simply preserving the momentum), and finally straightens out.

Now, the thing that surprised me most in the course of the investigation was a wikipedia reference to the 1969 article by T. Kane and M. P. Scher "A dynamical explanation of the falling cat phenomenon" (International Journal of Solids and Structures 5 (7): 663–670. doi:10.1016/0020-7683(69)90086-9), as the solution as "originally due to (Kane & Scher 1969)." This appears to imply that the problem remained (officially, at least) unsolved for about 80 long years - quite on a par with the, say, better known Poincaré conjecture. But think of it, most probably the members of the Académie des sciences in Paris were not the first to ponder the question, which leads to a conclusion that the question has a much longer history. Hmm, I would never guess.

Wikipedia notes that

The cat righting reflex is a cat's innate ability to orient itself as it falls in order to land on its feet. The righting reflex begins to appear at 3–4 weeks of age, and is perfected at 7 weeks.

This discovery is reported as late as 19 December 2011! (Disparagingly, the link to the research is not publicly available.) What is unclear is by which method the researcher came up with the "3–4 weeks of age" estimate and how big was the observed feline population.