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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?

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    $\begingroup$ It is not really clear what you are after, but the space $\mathbb R^2$ equipped with the distance $$d((x, y), (x', y'))=|x-x'|+|y-y'|$$ looks good. $\endgroup$ – Giuseppe Negro Sep 20 '18 at 7:07

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