Real Tennis Balls

A real tennis (distinct from lawn tennis) ball consists of a spherical core around which strips of cloth are wound by hand. Then there is a leather cover. Each and every ball is unique!

Winding the thread around the central core so that the entire surface is covered evenly in layers, so that as perfect a sphere as possible is formed, sounds to me like it should be a known and solved problem. Can anyone help with the solution?

Cheers

I certainly can't provide it, but I can try to help with it.

Instead of considering the thread itself, it's easier to consider the axis about which (at each moment) the thread is being wound. That is, the pair of points on the ball's surface, that are as far as possible from the recent course of the thread. We want these two points to move evenly over the surface of the sphere, so as to visit every region of it, and no region too often.

Instead of considering a pair of points moving over a sphere, we can consider one point moving over a projective plane, again so as to visit every region of it, and no region too often.

An obvious solution is to start somewhere and spiral outwards. However this won't work too well. We should prefer to have our point (on the pp) or pair of antipodal points (on the sphere) move fairly rapidly.

A good puzzle, I think.

Sure. It's got various names: - the hairy ball theorm - the hedgehog theorm etc. etc.

And the usual chant from the grad-students at the back of the seminar is 'you can't comb a hairy ball.'

Ok, that sounds silly. But basically it goes like so. If you try to construct a tangent vector field on a sphere, you will discover that at some point you come to a problem. There will be a part or a singularity or some kind of irregularity that makes the field not tangent.

So, consider your windings. Let the thread have a direction. If one thread covers a place then you have a tangent vector defined by a unit vector in that direction. If two, then the average of the two defines the tangent vector. If the cover were smooth and uniform, it would smoothly define a tangent vector field at every point on the surface. But that is impossible. So there is no perfectly smooth covering of the ball. At some point, you will have some kind of irregularity, a part or a nap or a thicker spot or something.

Basically, such balls are wound with thread thin enough that one extra thickness does not exceed the quality measures of the ball.