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Dan · Joined: Dec 31, 1969 Posts: 290 Location: USA

Topologists say Moebius is a surface with one side. No it's Not, it is a single surface with two sides as you can see for yourself = it objectively has two sides perpindicular to each other.

To Make A Moebius: take a long thin rectangle of paper like a one inch wide strip from an 11" letter size paper and rotate one end 180 degrees and tape or glue the two ends together. Equal a single continuous surface with one continuous edge and two sides.

Which implies that the two sides of the surface must be separated, or violate a very old scientific observation implied by postulating the universe is conserved: Two things can not occupy the same point/time in 3-d space/time.

Raises these two questions, 'What do topolgists mean by a surface?', and 'What do they mean by a side?'.

The answer must say something about where topologists put(unconsciously assume) the 'observer' = themselves sits with respect to the surface or side.

Seeing a side of something necessarily implies to me that there is another side.

But I can be on a surface but not 'see' it, i.e. leave my perception of being on it out of consideration(what I posit topolgists do). So I can go round a sphere on its surface to its other side implying an inside of the sphere(balloon) = another 'surface inside sphere' = other side of surface. thus two other sides; A. other side of sphere staying on 'out side sphere'; B. inside of sphere looking out at backside of outside sphere surface.

So it seems to me a surface = something I can go unimpeded across and return to my starting point without going through the surface = pushed around on it by a topologist.

While a side = a surface I must go through(see through) to get to the other side of surface.

Ergo: The difference between side and surface depends on the 'degrees of freedom of movement' of observer with respect to surface or side. Surface = go across or on; while side = go through or stop.

So lets do a thought experiment: I, the topologist, am on a surface and I sit like a "unit length stick" always perpindicular to the surface, and I can move in any direction across surface no matter how it bends and curves always staying perpindicular to surface. My vision is that of a laser that projects exactly at right angles to my stick in one direction. Ergo tangent to the surface where my 'stick body' touches surface.

Implication: Where the surface curves down away from my laser like eye, I see nothing but space, and where the surface curves up and intersects my laser like eye ray, "I see the side of the surface.".

In short the distinction is that "I am on a surface"; but "I see sides."; thus from an observers point of view, all surfaces have at least one other side. i.e. In the case of a sphere an observer can see from both an inside sphere looking out "other side of surface" and an outside of sphere looking through sphere "other side of surface of sphere" = two 'other sides.

e.g In the case of a moebius, we actually have four 'other sides' to perceive.
A. From sitting on outside surface of moebius = 1 side at a time;
B. Since, by conservation, there is a distance between the two sides of the moebius surface; then "from between the two sides of the moebius surface"
looking up = 1 side;
and C. looking down = 1 side; so I can potentially see both 'looking out from between sides of moebius surfaces at same time.
D. Potentially, looking at a moebius from way off outside of its surface, I can see two sides of surface at same time. e.g. our strip of paper with end rotated 180 eegrees and glued.

Implication: From between sides of moebius surface an observer can potentially see 'two sides' at same time; while from outside an observer can only see one side from on surface or close to it.

Implication: Where you sit = where you see moebius from must be EXPLICITLY included in your logical description of a mobius.

Obviously, mathemeticians have not done so before.

Since I am asserting that we live in a moebius strip shaped universe, answering the question: "How do get topologists to explicitly include themselves as observer in their descriptions of a moebius?" is crucial. I bet I beat them to it.

Ergo, We have a Discrepancy between what mathematicians say, and what observers can see. Lets have a public disagreement to resolve this question.