Mathematical possibilities of Newton's Third Law

 $$\textbf{F}(\textbf{r}_1,\textbf{r}_2)=\textbf{F}(\textbf{r}_2,\textbf{r}_1)$$

The simplest possibilities are 

$$r=|\textbf{r}_1-\textbf{r}_2|$$$$\textbf{r}=\textbf{r}_1+\textbf{r}_2$$

$$r=\textbf{r}_1\cdot\textbf{r}_2.$$

The law does not forbid them so they must occur in Nature between Fermions...

Now, what if $$\textbf{F}(\textbf{r}_1,\textbf{r}_2)=-\textbf{F}(\textbf{r}_2,\textbf{r}_1)?$$

Then the simplest possibilities are 

$$\textbf{r}=\textbf{r}_1-\textbf{r}_2,$$ and $$\textbf{r}=\textbf{r}_1\times\textbf{r}_2.$$

We suspect these are the kinds of forces that happen between Bosons...