By Ian Stewart
Falling cats can turn over in mid-air. Well, most cats can. Our first cat, Seamus, didn’t have a clue. My wife, worried he might fall off a fence and hurt himself, tried to train him by holding him over a cushion and letting go. He enjoyed the game, but he never learned how to flip himself over.
Maybe Seamus was puzzled for the same reason scientists were, a hundred years ago. On the face of it, the feat seems impossible. Flapping your paws and hoping that air resistance will do the trick is over-optimistic. Since the cat has nothing to push against, how can it make itself rotate?
In fact, there seems to be a good mathematical reason why it can’t be possible. It’s called the law of conservation of angular momentum, which is a fancy way to say that you can’t create spin from nothing. How can an upside-down cat turn over without spinning?
In 1894 a French doctor, Étienne Jules Marey, took a series of photos of a falling cat and discovered that it doesn’t. Spin, that is. But it does turn over. How come?
Now they had pictures to look at, the scientists figured it out. What really matters isn’t little bits of spin here and there. What can’t change is the overall spin of the entire cat. It starts with a total spin of zero: motionless. The same is true at the end. So the cat doesn’t have to create or lose any spin. It just has to wiggle various appendages so that the spin remains zero, but the cat flips over.
That wouldn’t be possible if the cat were a rigid body, but it’s actually very flexible. It can change its shape. And shape-changing is what makes the trick work. Some bits of the animal can turn one way, while other bits simultaneously turn the other way, keeping the overall spin at zero. Fit it all together, and the cat can flip over.
Here’s how. First, the cat sticks out its back legs and pulls in its front legs. Then it twists its rear end slightly one way, and twists its front end in the opposite direction. The total spin remains zero, but the front end twists a lot more than the rear because of the positions of the legs. Then the cat pulls in its back legs, sticks out its front legs, and twists everything back the way it came. However, the back end now twists more than the front, because of the changes in which legs stick out and which don’t. So neither end of the cat goes back to its original position.
The net effect is that the cat flips over, but the total spin at all stages is zero. The cat has to do all this in mid-air, while falling. I doubt it does the sums: the technique is instinctive. The ability must have evolved over millions of years. Somewhere along the line, Seamus missed out.
Ian Stewart is Emeritus Professor and Digital Media Fellow in the Mathematics Department at Warwick University, with special responsibility for public awareness of mathematics and science. He has written many books, including How to Cut a Cake and Cows in the Maze for OUP. This post first appeared on the BBC Focus/Oxford University Press microsite.