directed energy

[AntonioLao]

The idea of fiber bundle was originated from the idea of tangent bundle. That is given a multi-dimensional surface, the collective vector bundle of all tangents to every point. Note that each point has infinitely many tangents. So, infinity of point would have infinity times infinity of tangents. I don't like this type of mathematical argument because infinity is not a number as defined in abstract mathematics. Abstract math uses the concept of topological connectiveness. However, non-orientable topology can be used to define the property of infinite directions such that a quantized direction only has 8 distinct properties.

[Bogie]

The idea of fiber bundle was originated from the idea of tangent bundle. That is given a multi-dimensional surface, the collective vector bundle of all tangents to every point. Note that each point has infinitely many tangents. So, infinity of point would have infinity times infinity of tangents.

Yes it would .

I don't like this type of mathematical argument because infinity is not a number as defined in abstract mathematics. Abstract math uses the concept of topological connectiveness. However, non-orientable topology can be used to define the property of infinite directions such that a quantized direction only has 8 distinct properties.

It would seem very convenient to reduce the infinite number of tangents at any point down to eight. Are you approaching your project on the basis that nature can be predicted by the math so that if you can get the math to work there is at least a possibility that it describes reality?

[AntonioLao]

The patterns (shapes and sizes) of Nature are described by using rational numbers that is the comparison of one integer over another. However, if these patterns of nature require certain directional properties then no math is capable to completely describe them. On the other hand, given a non-orientable surface, the directional property of right-up-forward is distinctively different from the directional property of left-up-forward. For the three binary sets of orientable directions: left(L)-right(R), forward(F)-backward(B), and up(U)-down(D) there are only 8 possible distinct directional properties: LFU, LFD, LBU, LBD, RFU, RFD, RBU, and RBD. To be observable, a three dimensional object has to have all 8 properties.

[Bogie]

The patterns (shapes and sizes) of Nature are described by using rational numbers that is the comparison of one integer over another. However, if these patterns of nature require certain directional properties then no math is capable to completely describe them. On the other hand, given a non-orientable surface, the directional property of right-up-forward is distinctively different from the directional property of left-up-forward. For the three binary sets of orientable directions: left(L)-right(R), forward(F)-backward(B), and up(U)-down(D) there are only 8 possible distinct directional properties: LFU, LFD, LBU, LBD, RFU, RFD, RBU, and RBD. To be observable, a three dimensional object has to have all 8 properties.

Yes, I see what you mean. Would it be appropriate to say that each directional property divides the set of infinite tangents into a 1/8 th subset so that any tangent will fall in one of the eight subsets?

[AntonioLao]

divides the set of infinite tangents into a 1/8 th subset so that any tangent will fall in one of the eight subsets?

Since the Mobius-Hopf-Klein topology describes one-sided surfaces, the infinity is restricted to a one-sided surface such that its non-orientability allows a left vector to become a right vector after completing one cycle (360 degrees) and after two cycles the left vector becomes a right vector again. Therefore, the topology holds a 720 degrees directional invariance.

[Bogie]

Since the Mobius-Hopf-Klein topology describes one-sided surfaces, the infinity is restricted to a one-sided surface such that its non-orientability allows a left vector to become a right vector after completing one cycle (360 degrees) and after two cycles the left vector becomes a right vector again. Therefore, the topology holds a 720 degrees directional invariance.

So any given tangent in a normal coordinate system, say Euclidean, would be able to assume different directional properties in the mobius_Hopf_Klein topology?

[AntonioLao]

The general covariance of the topology makes the use of a coordinate system unnecessary and counterproductive.