Since points on a euclidean plane can be represented by one coordinate on a space-filling curve, is there any curve such that if two vectors $(x\_0,y\_0)$ and $(x\_1,y\_1)$ were represented by $a$ and $b$, the sum $a+b$ would represent the vector $(x\_0+x\_1,y\_0+y\_1)$? Could this curve be generalized to three dimensions?


EDIT: Even one with $ab$ representing $(x\_0+x\_1,y\_0+y\_1)$ would be a start, I can't find anything.


EDIT2: Never mind $a+b$, is there any way to do some sort of simple operation between $a$ and $b$ to represent vector addition?



Space filling curve to simplify vector addition?
There can be no continuous space-filling curve respecting $+$ in the way you desire. By considering $0+b$, it follows that $0$ must represent $(0,0)$. Now, suppose that some real number $a$ represents $(1,1)$. It follows that $2a$ represents $(2,2)$ and $\frac{1}{2}$ represents $(\frac12,\frac12)$, and in general, a rational number $q$ must represent $(q,q)$. Extending by continuity, the whole curve will lie on the diagonal line, a contradiction.


