A Geometric Mental Exercise

[KALSTER]

This is one of a few scenario's I was contemplating today 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 sphere 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 sphere 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 sphere gets. Every now and again a very large fluctuation might occur and the sphere would only be spherical on average. In fact, it could statistically form almost any shape 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 extend this sphere into infinity. Would shapes still be possible internally as density varies? I think sure.

 

 

 

EDIT: It has occurred to me that if only one of three dimensions are allowed that the skin enclosing the point sources would be a rough cube. Only if combination vectors are allowed would a sphere result. But then, how would the frequency and variety of possible geometric shapes be affected by either scenario in an infinite volume?

Edited July 22, 2008 by KALSTER
multiple post merged/ fixed error

[prometeu]

This is my geometric mental exercise: construct a three dimensional cube in a paper with two sides parallel to the paper, or a cylinder with the basis parallel to the paper. Of course one of the side is "close" to you. Now imagine that the other side is close to you. Change the sides alternately as fast as you can and as many times as you can. In some point you are incapable to do it just one more and you ask yorself what was I doing. Try it.

[Kyrisch]

This is one of a few scenario's I was contemplating today 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.

 

Okay, I'm following...

 

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?

 

Any reference point wouldn't change position, it would only oscillate with the shape just as a buoy rises up and over a wave. The buoy never changes displacement in the direction of motion of the wave.

 

Actually, the way I see it, the torus would oscillate such that any cross section would be eccentric to some degree and would only be circular at equilibrium.

 

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

 

Suppose you have a sphere 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 sphere 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 sphere gets. Every now and again a very large fluctuation might occur and the sphere would only be spherical on average. In fact, it could statistically form almost any shape 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 extend this sphere into infinity. Would shapes still be possible internally as density varies? I think sure.

 

 

 

EDIT: It has occurred to me that if only one of three dimensions are allowed that the skin enclosing the point sources would be a rough cube. Only if combination vectors are allowed would a sphere result. But then, how would the frequency and variety of possible geometric shapes be affected by either scenario in an infinite volume?

 

I dunno if any of this would work because a sphere encompasses an infinite amount of zero-dimensional points, not near-infinite. Would what you proprosed still work? You were talking about probability, that at any one time it is, say, 25% likely that a point will "stray", so in total, 25% of the enclosed points, at any one time, will be "straying". But what is 25% of infinity?

[KALSTER]

I dunno if any of this would work because a sphere encompasses an infinite amount of zero-dimensional points, not near-infinite. Would what you proprosed still work? You were talking about probability, that at any one time it is, say, 25% likely that a point will "stray", so in total, 25% of the enclosed points, at any one time, will be "straying". But what is 25% of infinity?

I had this problem on another forum as well. Let me provide a hopefully clearer discription. I agree with what you are saying. 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 you 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 you say, 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.

Edited July 25, 2008 by KALSTER