[AntonioLao]

It is through the mathematical theory of surfaces that allows the classifications of topological time invariance. These classifications are intrinsically performed with lesser dimension than the actual number of dimension. The key concept is the mathematical notion of ‘sides’. At the outset, it is important to bear in mind that mathematical surfaces do not have ‘sides’, but the concept of ‘sides’ can still be made, from a vantage point of higher dimension. For the surfaces, this point is the 3rd mathematical dimension (not the same as the 3rd physical dimension). For 2D creatures, the concept of ‘sides’ makes no sense, just as it makes no sense for us to talk about the boundary of our 3D physical universe. But if we, again, postulate the time dimension as the 4th physical dimension then observations made through 3D space and 1D time allow the distinction of past and future, and the present or ‘now’ becomes the ‘sides’ of our 3D universe viewed from 4D space-time. There is a fundamental subtle distinction between the space-time used above and the space-time of Einstein’s relativities. This distinction will now be discussed.

Einstein’s space-time is based on the ‘orientable’ cylindrical band (2-sided and topologically equivalent to the light-cones), while the other topologically distinct surface is the Moebius band. Moebius band is nonorientable (1-sided). For clarity, we now replaced the word ‘orientable’ with the word ‘directional’. So, for a 2-sided physical universe, there is distinction between past and future and time flows is unidirectional (always toward the future). And for all topologically equivalent closed surfaces to the sphere, time past is the inside and time future is the outside and time present of ‘now’ is the temporal boundary separating past and future.

But for a 1-sided physical universe, time flows is non-directional (no preferred direction). There is no distinction between past and future and the flow of time is random or the same as saying every possible direction is equally likely.

It is imperative that these two distinct topological flows of time must be combined in order to unify quantum mechanics and general relativity and the conviction that it can only be done in one physical dimension with a new principle of directional invariance. This principle is already discussed elsewhere in the forum.