Appreciate the attempt, but in this example the 4th axis is not independent since the motion along that axis can be achieved, with some complexity, by the motio

Appreciate the attempt, but in this example the 4th axis is not independent since the motion along that axis can be achieved, with some complexity, by the motion along the other axes. Granted this is not very useful for a machinist because it will be very tedious to machine a part this way compared to the dedicated 4th rotating axis, but mathematically it is redundant. I have found it easiest to think of a logical dimensions or configurations when thinking of higher dimensions. Physically it can be a row of bulbs (lighted or not) wherein N bulbs (dimensions) can represent 2^n states in total. The 2 here can be increased by having bulbs that can light up in many colours.