Now take the section determined by the zy plane. This will be a circle as shown in [fig. 45]. If we let drop the x axis, this circle is all we have of the sphere. Letting the w axis now run in the place of the old x axis we have the space yzw, and in this space all that we have of the sphere is the circle. Fig. 45 then represents all that there is of the sphere in the space of yzw. In this space it is evident that the rods CD and EF can turn round the circumference as an axis. If the matter of the spherical shell is sufficiently extensible to allow the particles C and E to become as widely separated as they would be in the positions D and F, then the strip of matter represented by CD and EF and a multitude of rods like them can turn round the circular circumference.
Thus this particular section of the sphere can turn inside out, and what holds for any one section holds for all. Hence in four dimensions the whole sphere can, if extensible turn inside out. Moreover, any part of it—a bowl-shaped portion, for instance—can turn inside out, and so on round and round.
This is really no more than we had before in the rotation about a plane, except that we see that the plane can, in the case of extensible matter, be curved, and still play the part of an axis.
If we suppose the spherical shell to be of four-dimensional matter, our representation will be a little different. Let us suppose there to be a small thickness to the matter in the fourth dimension. This would make no difference in [fig. 44], for that merely shows the view in the xyz space. But when the x axis is let drop, and the w axis comes in, then the rods CD and EF which represent the matter of the shell, will have a certain thickness perpendicular to the plane of the paper on which they are drawn. If they have a thickness in the fourth dimension they will show this thickness when looked at from the direction of the w axis.
Supposing these rods, then, to be small slabs strung on the circumference of the circle in [fig. 45], we see that there will not be in this case either any obstacle to their turning round the circumference. We can have a shell of extensible material or of fluid material turning inside out in four dimensions.
And we must remember that in four dimensions there is no such thing as rotation round an axis. If we want to investigate the motion of fluids in four dimensions we must take a movement about an axis in our space, and find the corresponding movement about a plane in four space.
Now, of all the movements which take place in fluids, the most important from a physical point of view is vortex motion.
A vortex is a whirl or eddy—it is shown in the gyrating wreaths of dust seen on a summer day; it is exhibited on a larger scale in the destructive march of a cyclone.
A wheel whirling round will throw off the water on it. But when this circling motion takes place in a liquid itself it is strangely persistent. There is, of course, a certain cohesion between the particles of water by which they mutually impede their motions. But in a liquid devoid of friction, such that every particle is free from lateral cohesion on its path of motion, it can be shown that a vortex or eddy separates from the mass of the fluid a certain portion, which always remain in that vortex.
The shape of the vortex may alter, but it always consists of the same particles of the fluid.