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..... tell me how far they are to your left/right, how far they are up/down, and how far they are front/back of your position. These are three numbers, and so we live in three spatial dimensions......

True, you can describe how to get from one object to another with only one number (kind of). You still have to give a direction, which is a vector consisting of three numbers. In other words, you could tell me "go 50 meters straight ahead to get to the sheepy farm." But by telling me that, you've implied that I go 0 meters left/right and 0 meters up/down.

One number would be a direction and the other number would be a distance in that direction. The direction number would define a direction superimposed on a spherical surface relative to the centre of the sphere. The combination of the two numbers would represent a sort of spatial vector.

The other way of looking at this is to think of each possible direction as a dimension, so you end up with as many dimensions as you need to obtain the required resolution, and a single number to represent the distance in that dimension.

Geezer - I am not sure we can define a position in 3-ball with two simple numbers.QuoteOne number would be a direction and the other number would be a distance in that direction. The direction number would define a direction superimposed on a spherical surface relative to the centre of the sphere. The combination of the two numbers would represent a sort of spatial vector. The only way I can see one number uniquely defining a direction from centre of sphere to the surface is if that surface has been divided and each section labelled with a number. As this numbering would be arbitrary I could not envisage how this could be used mathematically - with polar coordinated or cartesian coordinated we can determine distances and relations between points through their description, and these coordinate systems allow for transformation.

QuoteThe other way of looking at this is to think of each possible direction as a dimension, so you end up with as many dimensions as you need to obtain the required resolution, and a single number to represent the distance in that dimension. this bit I agree with entirely - a comboined unit vector and magnitude

Geezer - I am not sure we can define a position in 3-ball with two simple numbers.QuoteOne number would be a direction and the other number would be a distance in that direction. The direction number would define a direction superimposed on a spherical surface relative to the centre of the sphere. The combination of the two numbers would represent a sort of spatial vector. The only way I can see one number uniquely defining a direction from centre of sphere to the surface is if that surface has been divided and each section labelled with a number. As this numbering would be arbitrary I could not envisage how this could be used mathematically -

I'm with Matthew on this one. If you want to specify a position on the surface of a sphere with perfect resolution, you'd need two numbers. I think mathematically, the explanation is that if you're divide the surface into boxes, you can also arrange those boxes in a line, so you're actually in one dimension. If your surface is continuous, i.e. it has infinite resolution, you can't arrange it in a line.

There is no way to map a sphere to a line if the possible positions on the sphere are continuous, which appears to be the case for physical objects.

Quote from: JP on 30/10/2010 04:15:16There is no way to map a sphere to a line if the possible positions on the sphere are continuous, which appears to be the case for physical objects.Bbbbbbut the possible positions on a line are also continuous, so it follows that there is no way to map a point in space to a line.

I still contest that to get from A to B you need only travel in a straight line. I.E along one dimension.

If there are three dimensions then an object only has the option of going along one of three paths and if none of those paths leads to where it needs to be then would be forced to take a series of steps at right angles to each other. That would also mean that there would be areas in between these dimensions that nothing could ever get to.

The notion that we need three numbers to describe where two points are relative to each other also seems to be an illusion. This system only works if we use a third point as an origin. Without it all we can do is say how far apart the two objects are. Again this can be measured in a straight line and one dimension is sufficient. On a cosmic scale its impossible to now the actual position of anything relative to anything else anyway because everything in the universe is in motion and the universe itself is expanding. What point would we use as the origin?

In a theoretical one dimensional universe as described in flatland (it was reading this book that set me thinking) is it not possible for the one dimension to be curved? Even loop back on itself to form a circle? Or for that matter loop randomly around until it forms a kind of cotton wool ball that touches itself in every direction. In that kind of universe an object could move in any direction and still only travel along one dimension!