If we consider like Mike NS did a single dimension 2^1, this gives us a straight line of length one. 2^2 a square of area 4. 2^3 a cube of volume 8.

Here's the controversial bit

2^4=(2^2)^2

We have defined the fourth dimension in a two dimensional plane. You could just say this gives you nothing special, only a sqaure with four times the area of 2^2. So in response to Mike NS's topic I think that when trying to figure out these neat methods for defining the fourth dimension, we tend to end up simply getting larger versions of shapes in lower dimensions. We must all talk about this more. You all must see this link, even if its just for the pictures. Read George Gamow's One two three infinity or type supercube into google images. The image you will see is a projection, as though you had shone light onto, a four dimensional cube.

http://en.wikipedia.org/wiki/Image:D...dimensions.jpg