About Spheres and their Generalizations of Higher Dimensional Spaces

There is no branch of mathematics, however abstract, which may not some day be applied to phenomena of the real world.

— Nikolai Lobachevsky

If a math man were to be left alone, and devoid of the physics, would he understand what the physics world could impart if he were not tied to it in some way?:)

Poincaré Conjecture

If we stretch a rubber band around the surface of an apple, then we can shrink it down to a point by moving it slowly, without tearing it and without allowing it to leave the surface. On the other hand, if we imagine that the same rubber band has somehow been stretched in the appropriate direction around a doughnut, then there is no way of shrinking it to a point without breaking either the rubber band or the doughnut. We say the surface of the apple is "simply connected," but that the surface of the doughnut is not. Poincaré, almost a hundred years ago, knew that a two dimensional sphere is essentially characterized by this property of simple connectivity, and asked the corresponding question for the three dimensional sphere (the set of points in four dimensional space at unit distance from the origin). This question turned out to be extraordinarily difficult, and mathematicians have been struggling with it ever since.

Now it is of course with some understanding, that few would recognize what this conjecture to mean, and having just read Andre' Weil's last word I do not think it to shabby to say that in this case, the Poincare conjecture has been recognized as a valid conjecture, that has taken some time in it being resolved.




For me being presented with the cosmic string scenario might seem just as complex, when in considertaion of the brane. How our universe could be contained in it. Some might of laughed it off quite easily, being part of some revolution of strings to M-theory, that it could include 11 dimensions.How would you embed these dimensions with these shapes?

What fascinates me, is how we could have found such visualizations of these topological forms within the the brane world and how this may have been described?

So I am looking for traces of literature that would help me in this direction. For example, how a torus would be looked at in a 2 dimensional sheet. Would this be relevant to brane world happenings if we considered, the example of "sound" (make sure speakers are on) in higher dimensions, as viable means of expression of these curvatures of those same shapes?



Of course I know I have to explain myself here, and the intuitive jump I am making. Could be wrong?

Part of this struggle to comprehend what has happened with bringing GR and QM together in one consistent framework, was to undertand that you have altered the perceptions of what dimension will mean? To me, this says that in order now for us to percieve what the poincare conjecture might have implied, that we also understand the framework with which this will show itself, as we look at these forms.

The second panel I showed of the graph, and then the topological form beside it, made sense, in that this higher dimension shown in terms of brane world happenings would have revealled the torus as well in this mapping. This is of course speculation on my part and might fall in response to appropriate knowledge of our mathematical minds. This gives one a flavour of the idea of those extra dimensions.



Well part of this developement of thinking goes to what these three gentlemen have developed for us in our new conceptual frameworks of higher dimenions. If you ask what higher dimension might mean, then I am certain one would have to understand how this concept applies to our thinking world. Without it, these shapes of topological forms would not make sense. It seems this way for me anyway.:)



I hazard to think that John Baez and others, might think they have found the answer to this as well, in what they percieve of the mighty soccerball, and the fifth solid I espouse:)