All hinges on the definition of dimensions in the model and the real world. The ant walking around the orange may know about its local up and down, without realising that the straight line it is walking is not really straight but is curving gently forever downwards. It is too busy not veering to left or right to wonder about the meaning of its horizon. But the local 'down' and the horizon 'down' are the same dimension, the 3rd after#1) left-right and #2) front-back.
So as with our toroid, its invention by taking the cylinder of the rotating rubber belt, and joining the ends of its axis, is just the macro version of a dimension in which a piece of the cylinder exists in.
This is a trick, because the premise of discovering the 3rd dimension was that the ant lives in a 2D world, we have a "number of dimensions" ambiguity.
Having scotched pseudoscience, alchemy (sorry Sir Isaac), astrology and general expedient guessing, I still have to mention barmy ideas because they are the origins of out-of-the-box thinking. They include:
- Our development of logic has to put things in neat boxes, eg each dimension is distinct, whereas in reality the edges between one dimension and the next may be blurred. After all, matter and energy are connected by the shady wave-particle duality. And the computers of our brains are soft and sludgy, not a set of straight lines and switch nodes as our computers still are.
- Space as a medium, or "ether" was disproved by Newton and many after him, yet energy travels through 'something', and masses interact at a distance through nothing but space. The basis of my wondering about the shape of the universe is the dual impossibility of existence, and non-existence, and my barmy thought of the month (May 2009 AD) is that space is the realisation of the tension (I will think of a better word) between these two impossible conditions. (I keep recalling thhgttu "Improbability Drive" but I consider that to be more humour than sense, and my 'reconciliation of impossibilities' concept is independent and unconnected to good old Douglas Adams, bless him.)
- The rotation of space creating its distortion by the masses it is constantly accelerating, relates directly to the dimension of time. What a huge conundrum though, to our linear minds, to grasp the start and end of time as the same thing.
Yes these are pseudoscience (if that!), but alchemy led Newton towards real science so we shouldn't be knocking things out of court too soon.
I came to the toroid shape without knowing how cosmologists got to theirs. If the amateur model works in this case, maybe it can work in others. I need to find what cosmologists think space is - in terms that I might understand. But time - I don't expect anyone to have worked that out. It will have to wait till June . . .
Thursday 21 May 2009
Tuesday 12 May 2009
Jump to a familiar shape
The rubber belt or cylinder is a useful model to show how masses can appear to attract one another even though they don't. With the cylinder at rest and the masses scattered randomly around the inner surface, nothing attracts anything. But if the whole lot is spinning around the axis of the cylinder, then the masses, which as Newton found want to keep moving in a straight line, are being constantly forced by the elastic surface to follow its circular path, and because it is elastic, the surface is distorted, ie each mass makes its own bump in the sheet. Masses nearby each other have their own local bump, but there is also a combined effect where the place between their bumps is lower than the area outside them, and so they are pushed towards each other by the "outside" gradient being larger than the "inside" gradient. This can easily be understood whether the masses are similar or very different.
Newton found that the force of "attraction" followed the inverse square law, and it will be interesting to calculate how the bump gradients of the distorted elastic medium match up to the inverse square law.
The cylinder needs developing because (a) I think the total angular momentum should equal zero (b) the cylinder has an axis, presently a straight line (c) don't forget the 2D surface is our model of 3D space. We have to park (c) while we work on the others.
By joining the ends of the axis together it becomes finite but unbounded (necessary to avoid the impossibility of real infinity) though in so doing we have invented a new dimension (it is not the same dimension as the one we add to the 2D surface to restore 3D space). But joining the axis gives another possible benefit - is it possible that the resulting toroid, its surface rotating in the direction through its hole, has zero net angular momentum? Actually I doubt it, I need to get into toroids. Fascinating that the area of a torus (surface area of the toroid) is the same as the area of cylinder of same r and length equal to the large circumference.
Although I knew that cosmologists say the universe could be toroidal, I didn't know why, so this little discussion, with its convenient jumps, gives some ideas of how this shape can be arrived at.
Using some obviology we can put a boundary to one problem of the torus. The major axis of the torus (the line through the hole) is not relevant. This is because, if it was, we would have to join its ends (to avoid real infinity) but in a new dimension, creating a new torus, with its own major axis - we are back to "turtles all the way down". For my simple mind, one torus is all it takes. We don't have to worry about "what's outside the torus" - that is taken care of by the fact that the torus is only a 2D model, it represents 3D space.
So if we come back to the world we can see, progressing through time with the three familiar dimensions forward-back, left-right and head-feet, we can say that real space is finite but unbounded ie looking in any direction with our ultra-powerful telescope and waiting long enough we will see the back of our own head. The space we are looking through is not emptiness but has elastic properties like a rubber sheet, which is bent by resisting masses and is rotating in a way which puts a constant force on all masses yet conserves angular momentum.
I'd like to suggest that the missing "dark energy" is related to the net angular momentum of the universe system, and the associated work being done to accelerate all masses, of which the work done in gravitational attraction between masses is just a tiny fraction. Well, suggesting is easy . .
Newton found that the force of "attraction" followed the inverse square law, and it will be interesting to calculate how the bump gradients of the distorted elastic medium match up to the inverse square law.
The cylinder needs developing because (a) I think the total angular momentum should equal zero (b) the cylinder has an axis, presently a straight line (c) don't forget the 2D surface is our model of 3D space. We have to park (c) while we work on the others.
By joining the ends of the axis together it becomes finite but unbounded (necessary to avoid the impossibility of real infinity) though in so doing we have invented a new dimension (it is not the same dimension as the one we add to the 2D surface to restore 3D space). But joining the axis gives another possible benefit - is it possible that the resulting toroid, its surface rotating in the direction through its hole, has zero net angular momentum? Actually I doubt it, I need to get into toroids. Fascinating that the area of a torus (surface area of the toroid) is the same as the area of cylinder of same r and length equal to the large circumference.
Although I knew that cosmologists say the universe could be toroidal, I didn't know why, so this little discussion, with its convenient jumps, gives some ideas of how this shape can be arrived at.
Using some obviology we can put a boundary to one problem of the torus. The major axis of the torus (the line through the hole) is not relevant. This is because, if it was, we would have to join its ends (to avoid real infinity) but in a new dimension, creating a new torus, with its own major axis - we are back to "turtles all the way down". For my simple mind, one torus is all it takes. We don't have to worry about "what's outside the torus" - that is taken care of by the fact that the torus is only a 2D model, it represents 3D space.
So if we come back to the world we can see, progressing through time with the three familiar dimensions forward-back, left-right and head-feet, we can say that real space is finite but unbounded ie looking in any direction with our ultra-powerful telescope and waiting long enough we will see the back of our own head. The space we are looking through is not emptiness but has elastic properties like a rubber sheet, which is bent by resisting masses and is rotating in a way which puts a constant force on all masses yet conserves angular momentum.
I'd like to suggest that the missing "dark energy" is related to the net angular momentum of the universe system, and the associated work being done to accelerate all masses, of which the work done in gravitational attraction between masses is just a tiny fraction. Well, suggesting is easy . .
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