Projective geometry proved that parallel lines do meet a point at infinity. Riemann spherical geometry proved that there are no parallel lines and that all geodesics of all great circles meet at two arbitrary poles called the zenith and the nadir. Hyperbolic Lobachevskian geometry proved that a point not collinear with a given line, infinitely many parallel lines can be constructed passing through the point and parallel to the given line.

With these four distinctive geometries, the question can be asked do certain kinds of topology can be associated with each of them? Can a topological structure be shared among them? The answer is a ‘yes’ for the former and a ‘no’ for the latter. However, if the hyperbolic geometry is associated with a Mobius topology then sidetrack topology is created. A pair of parallel lines completing one cycle on the one-sided surface will become sidetracked such that the line on the left is now the line on the right, vice versa, right line is now left line. But another cycle put them back in their original position. Repeated cycles allow repeated switching of position. Nonetheless, in three dimensions there are two more degrees of freedom, switching between top and bottom and between forward and backward. Altogether there are 8 directional invariance properties to describe a 3D object, as required in formulating a quantum theory of the space-time continuum. These sidetrackings defined a physical concept called the frequency: temporal and spatial frequency or a combined spatio-temporal frequency as the quantum of the space-time continuum.