What is N-Dimensional geometry?

To find the answer you must first understand the theory of Parallel Co-ordinates:

Parallel Coordinates provide a method of visualising hundreds of variable interactions on one 2D graph. They are in essence a major scientific breakthrough and one which is harnessed by GPC technology.

 
Traditional graphs are restricted to a maximum of 3 variables. A single point plotted on a 3D graph in the traditional, orthogonal, manner is shown here...

 
The parallel coordinates method replaces the fundamental assumption of orthogonality between dimensions which has restricted previous co-ordinate systems to visualisation in a maximum of three dimensions. If dimensions (axes) are instead represented as parallel to each other then the limitation on their number (almost) disappears.

The 3 dimensional point shown above can be transformed into parallel coordinates. This is shown below.

Each axis has its own scale, and the datapoint is described as a polygonal line connecting the individual axis values. To recap: a point in cartesian coordinates is represented by a line in parallel coordinates.
 

Adding more dimensions to the parallel coordinate plot simply involves adding more axes to the right of the plot and extending the polygonal line to join up with the new points. It really is as simple as that.

For an example of an entire process using a GPC tool see the next diagram.All of the process values of variables P1 through P14 at approximately 0900 on the 14th of the month have been associated with their resulting lab qualities (q4 through q8) and are shown as one polygonal line.

Of course, a single point of operation will not tell us very much at all about our process.