From a person who does not fully understand topology. It seems to me that what Mobius and Listing both missed in their definition of the non-orientable surface of the so-called Mobius strip was the fact that it was non-orientable because it detracted from the underlying rules of geometry. For example, Mobius described a projective plane as that which may have within the plane numerous triangles which point in opposite directions when they share an edge. In this understanding, it is the triangle which forms the basis for the “non-orientable” surface. So, if we were to take a rectangle, draw some triangles on it, and then twist the edges to produce a mobius strip, we would find that the triangles do not follow the rule of having to point in opposite directions.
But here there is a problem. First, there is the presumption that the triangle is the standard for the orientable surface. Why did Mobius use the triangle for the standard of defining an orientable surface? This has to do with the definition of a polyhedra borrowed from Euler (v – e + f = 2). And this, of course, has its roots in Euclid’s Elements (definition 7 in book 1, on a “plane surface”). The triangle, though, is a plane surface and not a polyhedron, at base. Although I can not right now think it through it seems that this triangle has in some abstract sense an oedipal dimension of normalization. The oedipal delusion here consists of the belief that at the base of existence there is a trinitarian framework which is orientable.
The trinitarian framework is better understood as already being stretched across the non-orientable surface which includes the Symbolic, Imaginary, and Real, such that it is not flattened or dependent at base upon the imaginary dimension of the plane surface or the triangle. Thus, to describe the mobius strip as a plane surface which is cut and twisted upon itself misses the primordial dimension of the non-orientable trinity. The problem with the birth of the Mobius strip is that it began within the Imaginary or ‘geometral’ dimension, that is, within the imaginary dimension. Thus, even the folding of the rectangle itself was secondary to there already being a geometral plane. Thus, the discovery was itself imaginary.