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Here is a geometric formation of how a cat rotates when it falls.  Parts of the cat are represented by point particles.

The animation you enjoy here explains how cats do geometry. Do you know a tale of Sir Isaac Newton's cat? I think he or she
was happier than Scroedinger's anyway. I do not know Newton's cat was able to turn a somersault well, but Prof. I. Newton
seems not to have known the falling cat phenomenon is closely related with classical mechanics. It turnes out only recently that
the falling cat phenomenon can be explained in terms of classical mechanics and differential geometry. When a cat is put upside
down in the space and released without any force inputted, he cannot rotate his own body. That is, the motion of the cat after
being released is performed with vanishing angular momentum. Indeed, cats realize rotations with vanishing angular momentum.
It is a heavy task for us to draw a realistic cat, so that we have chosen to model the cat as a system of particles. This animation
shows vibrational motion with vanishing angular momentum. You will find that the first and the last configurations are the same.
That is, the vibration realizes a rotation as a result. We show by the connection theory that the realized rotation is interpreted as
a holonomy associated with a parallel translation. So cats unconsiously do geometry. (T.Iwai)
Remark:This work is part of research in our laboratory 1994.

I got this animation from