Isn't it just two ways of using the concept. A straight line has to have three dimensions to exist, but an idealised straight line has one dimension. Which you use depends on your purpose, mathematics, geometry or what might actually exist. If you look down the length of an idealised straight line you see a dimensionless point, but one which nevertheless has extension in the form of depth. That's my analogy for a fourth dimension, a co-ordinate system in 1D that extends as a straight line 'inwards' from each point in spacetime. This is possible (topographically) because all points on the surface of a sphere can be directly connected to the centre of the sphere by straight lines. We seem to agree with you that the sphere is more fundamental than its 3D surface. However I'm not sure yet that we're really talking about the same thing.
I'm not at all sure that I can explain what I mean about this. I know what I mean, but then I would wouldn't I. If spacetime is conceived of as existing on the surface of a sphere then the sphere must have one more dimension than that surface. If spacetime is 2D then the sphere must be 3D. If spacetime is 3D then the sphere must be 4D. Forgetting time and superstring loops for now then spacetime is 3D, and its expansion/contraction is uniform (aside from contingent conditions). It is therefore useful to model it as being the surface of a sphere which is expanding or contracting. This analogy works because geometrically it is possible for each point in 3D spacetime to be at right angles to the radii of the sphere and also for each point in spacetime to be the endpoint of a unique radius. Because of this the analogy works well, it has all the correct geometric properties to model the mathematics. I'm going further than this and suggesting that it works so well because the sphere actually exists and really does have those properties when seen from our 3D perspective. It is not really a sphere, it is a degree of freedom at each point in spacetime. We cannot visualise this very easily. However as the effect on spacetime of the existence of this extra degree of freedom is to give spacetime properties analogous to the surface of a sphere then it is helpful to think about it as being spherical. This is not the end of the story because this is just an explanation of why the analogy works, not a description of the real thing, which in my speculative theory is equally well modelled as a dimensionless point, an endlong view of an ideal straight line which is the radius of the sphere. But again that is a visualisation aid rather than a description. Another way I could say it is that as well as the three usual spatial dimensions there is 'in' and an 'out'. In a way this makes each point in spacetime something like a black hole (perhaps even exactly like one), a dimensionless point when viewed end on. Any collection of points behaves geometrically like a surface and is better modelled as a sphere. Thus when I say the sphere exists I don't mean that a great big expanding sphere in the fourth dimension (or leeaus's first) actually exists. I mean that the dimension that causes the sphere/point analogy to work actually exists. Any better?
Hello again King of Denmark All points on the surface of a sphere cannot be connected to its centre. That is a fundamental geometric misapprehension. A sphere does not have a centre. It has an infinite convergence towards a non existent centre. This, according to some, is the reason the universe exists. You would have seen such on that web page. Regards to your quest leeaus
It's all the same to me mate. All points on the surface are in contact with the volume that lies within the surface. That's sufficient for the analogy to work. Denmark? I have problems with the style of the web page and haven't got through it all yet.
KOD. Centre of a sphere and interior of a sphere. Both the same to you OK but the two aren't not considerable as the same. One exists, one doesn't. Your quest forever, mate. leeaus
Canute was an all conquering Viking warrior, Canute, is where the KOD comes in. Please enlighten on problems of style with web page. The objective was to communciate. leeaus
Of course he was. I'm an idiot. I was just thinking of him as King of England, I forgot his past. I'll get back with something specific about the site.
Leeaus - just went and had another look. Personal comment but I find your writing is too imprecise to convey your reasoning. I understand some of what you're getting at, but you don't really make your case, and it's hard work trying to follow it. That's not a criticism of your ideas by the way. I find it hard to be sure what your ideas are from your explanation of them. (I have the same problem, in fact most people do. It's damn hard work expressing something clearly).
Thanks for the comment Canute. The diagrams are supposed to do the talking. Perhaps the whole point of the page is there is nothing considerable as precise except the sphere. That all other attempts of precise ideas are opinions between humans. Just as 1 + 1 = 2 is ultimately only a guess. Never sure how you can scale a size that encompasses the universe onto a spread that the human eyes can look at. Infinite distance couldn’t begin at one end is where the idea comes from. If it did the distance before the beginning would not be accounted for within infinite distance. Therefore infinite distance has to begin inside its self. When you satisfy your self with that intellectual appraisal, the next question is that of zero distance. Zero distance in geometric language is the smallest sphere so its get pretty straight forward from there that infinite distance and zero distance are relative spheres. Would suggest that most begin any consideration of infinite distance at one end and then try and picture a line going forever. Once again the distance before the end is not accounted for in this consideration. And zero distance, well that is the convergence. Zero distance can only be approached, sort of like a frog halving his jumps never gets across the room. He jumps forever. A slightly different diagram is the following link. May help you see what is being got at. We or the universe is what is going on inside infinite distance, infinite distance being the circumferential length of the largest sphere. http://home.iprimus.com.au/siewk/convergence/ leeaus
leeaus - I agree with some of whay you suggest, but for more philosphical reasons. For instance when you say - "Therefore infinite distance has to begin inside its self" I see that as true, but oddly expressed. I would say that everything must 'begin inside itself' because that is what the Cosmos itself must invevitably do. Unless existence is ultmately entirely self-referential it wouldn't be here. The alternative is a 'first cause' and that doesn't make sense to me for it begs an unanswerable question.
Canute Absence of a beginning and an absence of an end may be the un-odd way of expressing what infinite distance must be. A piece of wood has a beginning and an end, you can locate places where the piece of wood isn’t. Should have written “begins within itself” perhaps. What was being expressed is that is where the logic personally began, that you could not attempt to comprehend infinite distance by gazing forth in one direction as far as you could. When you move onto the next logical step of infinite distance only being observable from within infinite distance, it becomes apparent that infinite distance geometrically is the shape of the sphere. Perhaps it serves no purpose to say so but at this point we have a deviation from your “everything must begin inside its self” suggestion. That is the shape of the sphere does not relate to time as the word “begin” suggests. The shape of the sphere begs no “first cause” as you put it. If you want to apply time logic to it, always exists would be the logic. Beyond measurement by time concepts is perhaps better. Take the universe out of the sphere if you like and you still have the shape of the sphere. An indestructible three dimensional structure. This sphere would be more light years than we can comprehend in diameter. Those diagrams of the sphere that have been provided are scaled down from a billions of light years diameter to several centimetres. If you care to consider as many characteristics of the shape of the sphere as you can, see how you go with one billions of light years across. Regards Leeaus
No not really Canute, just was ill at ease with “everything must begin inside its self” as stated by you. That’s all. Regards leeaus
It's a clumsy way of putting it I agree. I was following your comment that "Therefore infinite distance has to begin inside its self".
