Dimensions

A point has no dimension until it is moved to create a line. so:

1. A point becomes a line if it undergoes a translation into one dimension only;
2. A line becomes a plane if it undergoes a translation in a further new dimension;
3. A plane becomes a space if it undergoes a translation in a further new dimension; and
4. A space becomes a space-time manifold (surely there's a better name?) if it undergoes a translation in a further new dimension.

This appears to be true for the four 'obvious' dimensions. But my limited understanding of relativity is that space-time becomes warped by gravity. Now if we 'warp' a line, we have to move it in a new dimension from the single dimension in which it exists 'unwarped'.

So if space-time becomes warped, is it warped within the existing four dimensions or in a new fifth dimension? As far as I can see if space-time is the whole universe then any warping must be outside of the four dimension space and time dimensions so it must be a new fifth dimension. If the one dimensioal line is distorted outside of its one dimension this must be into a further dimension. Likwise for two, three or four dimensions.

So is gravity a fifth dimension?