A cylinder is a surface that can exist in any space with a notion of distance, as a 2 dimensional set of all points a certain distance away from a particular straight line. In a 3-sphere or hypersphere, which is a 3D surface of a 4D ball, a straight line is a great circle, a circle whose radius is equal to that of the sphere itself. This means that a cylinder within a 3-sphere is a torus. The space left over in the 3-sphere appears, with the help of stereographic projection, to be of the same shape, and with the right radius, the torus-cylinder divides the hypersphere into two identical doughnut/bagel—shaped pieces which are topologically interlinked and identical. This means it is possible to tile the 3-sphere with two identical, non-seperable tiles, unlike the 2-sphere, where this seems not to be possible (for example, the two regions into which the surface of a tennis ball is divided by the white line are identical but not interlinked) .
An interesting mathematical fact:
A cylinder is a surface that can exist in any space with a notion of distance, as a 2 dimensional set of all points a certain distance away from a particular straight line. In a 3-sphere or hypersphere, which is a 3D surface of a 4D ball, a straight line is a great circle, a circle whose radius is equal to that of the sphere itself. This means that a cylinder within a 3-sphere is a torus. The space left over in the 3-sphere appears, with the help of stereographic projection, to be of the same shape, and with the right radius, the torus-cylinder divides the hypersphere into two identical doughnut/bagel—shaped pieces which are topologically interlinked and identical. This means it is possible to tile the 3-sphere with two identical, non-seperable tiles, unlike the 2-sphere, where this seems not to be possible (for example, the two regions into which the surface of a tennis ball is divided by the white line are identical but not interlinked) .
Apparently this is related to a Reeb foliation .