© Spectrum of Science, Manon Bischoff (detail)
Wind direction north versus south pole
With this statement you are almost there! The wind direction can only turn by an integer multiple of 360 degrees along a circular route, otherwise the wind would have to blow in different directions at the start and end points (which are the same). Now one only has to realize what it means when a vector field is continuous: It must not change its orientation abruptly. So if the wind shifts exactly once while you are walking a certain circular path, then this is also the case if you deviate a little from the path you have taken. In fact, the number of complete rotations of the wind must be the same for each closed path. If this is not the case, then either the vector field is not continuous or it is zero at some point.
In our example, the wind direction varies by the same amount along the northern and southern polar circles, but with different signs: In the first case, the wind direction rotates clockwise, in the second counterclockwise. That is, once it rotates 360 degrees and once -360 degrees – which makes an angular difference of 720 degrees. So if the vector field is to be continuous, it must be zero at at least one point. Such a point is usually equivalent to a vortex in a continuous vector field.
The hedgehog theorem for the ball
From a meteorological point of view, this means that there is always a hurricane somewhere in the world whose eye is completely calm. However, it can be shown that this is not the case for a donut surface. Because, as already mentioned, a sphere and a torus (that’s the mathematical term) are drastically different from each other. If there were a toroidal planet, then the wind could blow anywhere without necessarily creating a hurricane anywhere.
The hedgehog theorem for the torus
In fact, the sentence has other practical implications that go beyond just combing your hair. Nuclear fusion, for example, requires strong forces that can withstand the pressure of a heated plasma. Since there are currently no suitable materials for this, one approach is to hold the particles together using a strong magnetic field. A spherical structure would initially be the most obvious, but because of the hedgehog theorem, the magnetic field is necessarily zero at one point – so the particles could escape there. That’s why the plasma experiments have a donut-shaped structure.