Geezer - I am not sure we can define a position in 3-ball with two simple numbers.

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 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.

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.

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.

this bit I agree with entirely - a comboined unit vector and magnitude

Yes, but you can't think of each possible direction as a dimension unless you're using the term very loosely.

There's a precise definition, but it takes some effort:

**A vector** is an ordered set of numbers, for example (1,2,3) is a vector, as is (.333,0,Pi) and (0,0,0) and (1,1,1), etc.

**A linear combination** of vectors is the result of adding vectors together, where each vector in the sum can be multiplied by a number, for example: 3x(1,2,3)+.2x(0,1,0)+10000x(0,0,0)=(3,6.2,9)

**A linearly independent set** of vectors is a collection of vectors such that no single vector can be written as a linear combination of the others in that collection, for example: (1,0,0),(0,1,0),(0,0,1).

Once you have these definitions, the dimensionality of your space is given by the maximum number of linearly independent vectors in your space that you can put in a set.

Phew!

The gist of this is, though, that any vector (a,b,c) can be written as a(1,0,0)+b(0,1,0)+c(0,0,1), so that by specifying a,b and c, you specify any vector in 3D space--that's why you can loosely say that the number of dimensions is given by the minimum set of numbers you need to describe a position in space.