How to reverse a sphere in a hypersphere

What happens?

To describe the hypersphere, the 4D analogon to the sphere, we can think of two solid tori that are glued together in such a way that the meridians on the surface of one torus coincide with the parallels on the surface of the other one, and viceversa. In the animation meridians and parallels are painted with different colours to emphasize the identification.

A sphere, painted blue outside and pink inside, is in the hypersurface of the hypersphere, inside one torus. We inflate the sphere till it touches the surface of the torus in a meridian. Since the surfaces of the tori are identified, the sphere appears also on the other torus, along a parallel.

We inflate the sphere further and the part on the surface of the tori passes entirely inside the second torus, till the sphere is represented by two disks (in the first torus) which are adjacent to an annulus (inside the second torus).

Now we can deflate the sphere symmetrically and it returns inside the torus where it was at the beginning, but painted pink outside. In other words, it is reversed: the inside surface is now outside and viceversa.

Instructions

Using the slider you can reverse a sphere by inflating and deflating it in a hypersphere, which is represented by two solid tori.

The buttonshows the linked tori, while clickingthe tori are drawn separate.

You can rotate the figure dragging the mouse with the left button down. Dragging the mouse with the right button down you can zoom to see the figure from a closer or further viewpoint.

The buttonstarts the animation, while the button pauses it.

You can reset the view to see the figure in the initial position clicking the button.