The dimensions of a set of three axes can be arranged in two ways; left or right handed. Cartesian co-ordinates are by convention always oriented to comply with the right-hand rule. It would seem this rule can be thought of as a cyclic transformation of order 3 which takes us from one axis to the next.
What are the analogues for left- and right- handedness in higher dimensions? Particularly in infinite dimensions?
We know that three is geometrically special arising out of the parallelisability of three-sphere, and the only higher dimensional space in which this happens again is 7-dimensions so is there only an analogue in 7-dimensions or can the concept be extended to other spaces?
In particular... and this is just a bit of background perhaps not material to the question. I'm interested in a space I'm constructing to study number theory in which every axis represents a prime number and every point along that axis represents an increment in the power of that prime number, so along the first axis we have 2, 4, 8, ... and on the 2nd axis we have 3, 9, 27, ... By this means every co-ordinate in the infinite-dimensional space represents a unique natural number given by the product of its co-ordinates. If the points along each axis are then spaced according to their square root, Pythagorus theorem guarantees that all points in the infinite-dimensional space are well-ordered by their distance from the origin, which is equal to the square root of the log of natural number they represent.
What I want to bring some understanding to, is how the process of counting in this space is described by some translation from any $x$ to $x+1$, and whether there might be some sense to be made of the rotation between axes that takes place with each increment.