Dimensions

I've only gotten as far as reasoning out that the 4th dimension is made up of an infinite number of 3D shapes.

Well that's basically deduction from the lower dimensions, isn't it? Infinite points make a line, infinite lines make a surface, infinite surfaces make a cube, infinite cubes.. is like infinite points, just bigger? No, you have to line the cubes up in the new direction to make a 4D entity, the picture in my head, no matter what I do, is still grouping them together in a 3D space.

Plane of existence sounds more like the "alternate world" thing than a spacial direction to me. Surely you can move from Inner, Ethereal, Material, Astral, Outer Planes, to the Far Realm and backwards, but up to 3D, hasn't space been infinitely large?