A Geometric Mental Exercise

[KALSTER]

This is one of a few scenario's I was contemplating yesterday while on a trip.

 

Suppose you have a square sheet of rubber. This rubber has zero internal friction and perfect elasticity. Now, take two sides and join them, forming a cylinder. Take one end and start rolling it up so that you are turning it inside-out. When you have done a bit, give it a final yank and let go. This is done in a perfect vacuum with no external interference.

 

What will happen?

 

As I see it, it should keep on rolling until it is completely inside-out. But then I think it should continue (as a result of momentum) and keep turning itself inside-out perpetually. The initial energy applied can't dissipate, so it has to keep on moving.

 

Now, imagine that in one oscillation, when the two open ends meet they fuse together, forming a torus.

 

What will happen to the motion then? I think a reference dot made on the inside of the torus should then move around in a circular motion around the outside and back to its original position, no?

 

 

 

Now on to a second and slightly more complex thought I had.

 

Suppose you have a skin enclosing a near infinite number of zero dimensional points. It would then simply be a single zero dimensional point. But let's say that each point can stray into any one of three dimensions at any time, forming a one dimensional string and then reverting back through the origin and into another dimension. The dimension it strays into is completely random. The degree to which it fluctuates, though, is determined by a probability curve. The smaller the fluctuation, the more likely it is. This probability curve might look something like a hyperbola, but with the symmetry being along the Y axis. The X axis would then be the vector degree of fluctuation (vector, as in it can fluctuate in any of two directions from the origin for each dimension) and the Y axis would be the frequency. Now, how would the volume look and behave now? It's size would be determined by the shape of the probability curve. The more likely larger fluctuations become, the bigger the volume gets. When only three dimensions are possible, the volume should average out into a cube, and if combination vectors are possible, it should average out into a sphere. If combination vectors are possible, it could statistically form almost any shape for an instant, given enough time. It could even form all kinds of shapes, or geometries, on the inside with varying density.

 

So how would the point sources fit together when the fluctuations occur? Does there have to be spaces in between? Would it still be able to have volume with no empty spaces in between? Each formed string should push away any adjacent strings, creating volume, no?

 

Now, what if we had an infinite number of point sources. Would shapes still be possible internally as density varies? How would the possible variety and frequency of geometries change between: (1)Only one of three vectors are possible at any one time, or (2)Combination vectors are possible?

 

What do you guys think?

[KALSTER]

Ok, let's stop the discussion on the rubber. It was just one of two things and the other thing is much more interesting to me.

 

Let me provide a hopefully clearer discription of the second one. I said a finite number of point sources, meaning point sources that have the added attributes as I discribed. If the construct were to be able to exhibit volume, then starting with an infinite number of point sources would negate the role the sphere/bag/skin plays in the setup, which is that one could form an intuative picture of volume created by the construct.

 

I wanted to set up the experiment in my mind with some added particulars and then see what happens. Points are, as has been said, zero dimensional. Lines are one dimensional. In physics a string is defined as a vibrating one dimensional line. I wanted to add the vibrating property of the strings into the construct later on, only after I have formed a complete mental picture of what is happening.

 

Anyway, the points I am talking about do not physically exist, only as the point of origin through which the physical one dimensional lines fluctuate.

 

These fluctuations occur roughly according to this graph:

 

 

As you can see from the graph, the chance of the strings being smaller increases substantially the smaller they get. In fact, one could describe the limit where the deviation from zero tends towards zero. So large deviation become unlikely to the extreme quite quickly.

 

That is, they can go in any direction and can elongate to any length, but with the constraint that they are more likely to be small than large. Let me make the speed at which they elongate, arbitrarily, the speed of light. So then my question was if this setup could exhibit volume. A point source will, over a sufficient period of time, form the rough appearance of a sphere. I am just wondering if, since the lines are only one dimensional, if a confined finite number or an infinite number would be able to affect each other, or “push” against each other. If the answer to this were to be no, that is when I would have to introduce the extra condition of the lines/strings vibrating (as proposed in current string theories). That would provide a measure of volume to each string, but it would also then force the necessity for gaps to form, that is, areas in the volume that is not occupied by anything at all. I was trying to avoid these gaps, for reasons to be discussed later.

 

You see, I am trying to consider candidate constructs for the space-time fabric, of which this one seems the most promising to date. At the moment I am thinking about whether the formed strings need to vibrate in order for the construct to be able to exhibit volume. The variation of two variables I can identify can then be responsible for inflation, namely the amount of vibration of the strings and the frequency distribution of longer deviations from zero of the strings.

[luc]

Well I don't think the vibration would be necessary. Imagine an infinite amount of strings forming a ball, they could do this without vibrating. An infinite amount of those balls could form any shape, with any density. However this would make it likely that gaps would form, but you could always imagine that those gaps are in turn filled with an infinite amount of balls, and so on.

 

EDIT: Now that I think further, The strings don't even need to form a whole ball, they just need to be oriented in such a way that they are a basis for 3 dimensional space. Then any space can be filled to a point where it approaches a solid volume, which isn't even the case in reality.

[KALSTER]

Well I don't think the vibration would be necessary. Imagine an infinite amount of strings forming a ball, they could do this without vibrating. An infinite amount of those balls could form any shape, with any density. However this would make it likely that gaps would form, but you could always imagine that those gaps are in turn filled with an infinite amount of balls, and so on.

 

EDIT: Now that I think further, The strings don't even need to form a whole ball, they just need to be oriented in such a way that they are a basis for 3 dimensional space. Then any space can be filled to a point where it approaches a solid volume, which isn't even the case in reality.

My misgivings regarding whether non-vibrating strings can exhibit volume is that a string only has a value in one dimension. That means that only when you lay them out end to end would anything add up. Simply laying them on top of each other would leave you with still only one string. I think that might be why branes are needed. A vibrating string though does exhibit an average 3D volume I think. Think of it like this. Lets say you stack a bunch of logs together, letting them touch each other on four sides of the circumference, stacking them into a cube. Now, if you lessen the circumference of each log, the size of the cube will shrink. If you continue to lessen the circumference until it reaches zero, the cube will have disappeared into a cross made by just two strings. If you then imagine an infinite volume of voluminous logs stacked similarly, you would have an infinite volume until the volume of each log is zero, after which you would be left with zero volume. So any tiny vibration of any kind would immediately have an infinite volume as result. IMHO