I know what sonic.blade means, though I'm not sure how well I'll go with explaining it - I haven't done maths of any real sort in years, and multi-dimensional stuff in even longer.
If we first of all think about a unit sphere. The sphere exists for values of x, y, and z between -1 and 1, and does not exist for all other values (outside the function's domain, as it were).
This is a simple concept to grasp, as we deal with three dimensions every day.
If you want to easily comprehend a 4th dimensional unit sphere, the best way is to arbitrarily assign some value to the 4th dimension. Most commonly, people assign time to the 4th dimension.
You can then imagine that at time t=0, a 4th dimensional unit sphere would look identical to a 3d unit sphere. But if you went back to t=-0.5 (keeping x, y and z at 1, for simplicity's sake), you would see a smaller sphere.
Well, if you think about a 3d sphere, a 2d cross-section taken at z=-0.5 is a smaller circle than at z=0. So for a 4d sphere, the "cross-section" at t=-0.5 will be a smaller sphere than that taken at t=0.
You can progress this across the entire domain of t (-1 to 1). If you remember that t is time, you can therefore imagine that 4d sphere would appear as nothing before t=-1, at which point it would become an infintesimal sphere. This sphere would grow until it reached unit dimensions at t=0, and would then shrink back to nothingness. Of course, this is just for visualising a 4d shape.
The elegance of this method means that you can assign practically any continuous property to a dimension. You can even use non-continuous properties, provided you limit the domain appropriately.
For example, you can set colour as an example. Colour is actually a value on an electromagnetic frequency spectrum. So if you assign colour to the 5th dimension, and set "red" to c=-1, and "blue" to c=1, with a smooth spectrum in between, you can imagine a 5th dimensional sphere.
This would be a growing/shrinking sphere, as for the 4th dimensional sphere, but the sphere would have an infinite number of spheres inside it, of differing colours. In the center of the 5-hypersphere would be two infintesimal spheres, one red, and one blue. The outside of the sphere when it was at it's maximum size and colour variation (x,y,z,t and c all equal 0) would be green. There would be a smooth gradient opf colour within the sphere, in BOTH directions (green->red and green->blue). Sort of like an everlasting gobstopper...
You can extend this however you like. Assign pitch, yaw and roll to dimensions - that brings us up to 8-d. Assign other properties - luminescence, transparency, roughness... brings us up to 11-d. You can even use non-visual properties (as these properties are simply there to help us understand the shape) such as volume, frequency of a generated noise, etc.
Maybe I've explained this poorly... I don't know... it's been too long.
Eighty-three percent of all statistical quotes are made up on the spot.