To be more specific, is there a way to generate a space in which
c = a + b
Where c is the shortest path joining the start of a to the end of b, but a and b are at a right angle to each other?
The way I imagined this was: Let there be a triangle with a right angle. Take the hypotenuse. Is it possible that, even though this hypotenuse is at an angle from both a and b (sides of this triangle), its length is actually always equal to the sum of lengths a and b ?
Basically this is a "what if Pythagoras theorem was actually c = a + b instead of the usual sum of the squares".
How would you define a transformation from this space (if possible) to our dear Euclidian space?