Rims and the GCD:For all rims {$\langle a,b,c\rangle$}, {$\frac{N(\langle a,b,c\rangle)}2+2=x_{_T}+y_{_T}+1$}. Proof: Let {$k$} be least such that {$\Delta^k(\langle a,b,c\rangle )=\langle y_{_T},0,0\rangle $}. Then {$\Delta^{k-1}(\langle a,b,c\rangle )=\langle y_{_T},y_{_T},0\rangle $}. By {$\Delta$} Respects Loops, {$N(\langle a,b,c\rangle=\langle y_{_T},y_{_T},0\rangle $}. A loop on the {$y_{_T}\times y_{_T}\times 0$} box generates a loop on the {$y_{_T}\times y_{_T}$} rectangle (seen from above). But two different loops on {$y_{_T}\times y_{_T}\times 0$} box generate the same loop on {$y_{_T}\times y_{_T}$} (even though the height of the box is zero, the top is still different from the bottom). By Billiard Loops and the GCD there are {$(y_{_T},y_{_T})-1=y_{_T}-1$} loops on the rectangle, hence {$2y_{_T}-2$} loops on the box. Then {$\frac{N(\langle a,b,c\rangle)}2+2=\frac12(2y_{_T}-2)+2=y_{_T}+1$}, which is {$x_{_T}+y_{_T}+1$} when {$x_{_T}=0$}. |