Comb one hairy ball It’s impossible, mathematics tells us! And there is really a theorem: the theorem Of the hairy ball, demonstrated in 1912 by the Dutch mathematician Luitzen Brower. Said in pop terms, this means that if we try to comb a hairy ball, the hair must necessarily overlap or form clearings, And we never manage to obtain uniform hairstyle. In this article we explain the theorem starting from the example of the hairy rugs and, passing from the example of the wheat fields and the hairy circle, we will clarify what the words have to do vector field And continuous with hairy balls. Finally, we will see how this theorem finds an application very concrete in the realization of the nuclear fusion.
The hairy ball theorem explained with hairy rugs
To understand what this curious theorem tells us, we start from the easy case of a hairy carpet: We can comb it in many different ways, as in the figure below.
In this case, the hairstyle at the top right and the one at the bottom left are uniforms, Without sudden changes of direction: each hair has almost the same direction as the hair that is close to it. On the contrary, in the first carpet at the top left the hair is combed from an outward point and some hair around that point go in opposite directions even if they are very close to each other. In the fourth carpet, at the bottom right, instead you can see a kind of small central clearing – What we would call a rose in hair – in which there seems to be no hair.
With the rugs we can make a little bit all the hairstyles we want, but if we try to comb hairy objects of different shapes then things get complicated.
In the figure above, the donut on the left is well combed so uniform, without clearings or other strange things. There ball, Instead, it has one at the top Little clearing: Is it possible to delete it by changing hairstyle? In fact, no! The hairy ball theorem, in fact, translated into the language of the “hairy balls” says that a uniform hairstyle can never be obtained on a ball: as far as we commit ourselves we will always find ourselves with some point in which clearings or other strange things are created.
Why do mathematicians speak of hairy balls?
But do the mathematicians really talk about combing hairy balls and hairy donuts? In fact they use these terms like colloquial jargon but they are not interested in the object “hairy ball” in itself, but speaking of hair, refer to the abstract mathematical concept of vector field. Let’s briefly see what it is.
The word field It is not found there by chance. For example, if we consider a field of wheat, we can think of every plant that appears from the ground like an arrow with its own length And direction: Each point of the ground has its own arrowjust like in the image under where the plants seem comb to form a drawing.
The idea that in Matematica is expressed with the concept of vector fieldit is precisely to attack a geometric object at each point of a geometric object arrow call vector.
What does “comb” fields mean?
Mathematics attack vector fields in any form, for example in the figure below we attacked four different Vector fields to a circumference, but only one of these concerns the hairy ball theorem, that continuous and tangent.
Said in simple words
a vector field And continuous If it does not present abrupt changes between arrows that are close to each other, but the changes must be gradual.
In the image above, at the top left, if we slide the arrow along the circumference We come across sudden changes of direction, so this is not a continuous field. In the second case, at the top right, the directions change in a way gradual, But there are some abrupt changes of length: this field is not continuous too. In the case at the bottom left, however, we witness gradual changes of both the directions and the lengths of arrows: this field is continuous. The last field is also continuous, but the arrows are all tangent to the circumference, this is what mathematicians would call comb. In essence, we combed a hairy circumference, but we did not deny the hairy ball theorem since this is a circumference, not a sphere.
What does the hairy ball theorem have to do with nuclear reactors?
A’application very concrete of this theorem it concerns construction of the reactors for the controlled nuclear fusion, like that of the project Process. Without going too much in detail, when these reactors are in operation they contain a plasma 10 times hottest than the sunbut at that temperature the material of no container could resist. The plasma particles, then, are confined using magnetic fields that force them to move in a closed space without them being in contact with any material. If these magnetic fields canceled, even if only in one point, the plasma could escape: but the hairy ball theorem tells us that this is exactly what would happen if spherical magnetic fields were used.
For this reason they prefer reactors shaped donut, called Tokamak: In short, for nuclear fusion a hairy donut can be better than a hairy ball … and the hairy ball theorem tells us.
