The Greatest Common Divisor of a BoxRelatively Prime Boxes made the link between a generalized Euclidean algorithm and loops. The link would be stronger if we could connect the numbers we get from the algorithm, {$x_{_T}$}, {$y_{_T}$} and {$z_{_T}$}, with the number of loops on the box. The link is there with rectangles. In Billiard Loops and the GCD we saw that the number of loops plus 1 is the greatest common divisor of the dimensions. Definition: Let {$N(\langle a,b,c\rangle )$} be the number of loops on the box with dimensions {$a$}, {$b$}, and {$c$}. {$\Delta$} Respects Loops: For all {$\langle a,b,c\rangle \in\mathcal {B}^+$}, {$N(\langle a,b,c\rangle )=N(\langle a,b,(a+b+c)\rangle )$}. {$N$} is Even: For all {$T\in\mathcal {B}^+$}, {$N(T)$} is an even number. In view of {$N$} is Even, we should look at {$\frac {N}2$} instead of {$N$}. If we think of {$x_{_T}, y_{_T}$} or {$x_{_T}, z_{_T}$} as representing the greatest common divisor of the box {$T$}, we should look for a relationship with {$\frac {N}2$}. We found one. It depends, though, on the classification of the box: Loops and the GCD: For all {$T\in\mathcal {B}^+$}, {$$\frac{N(\langle a,b,c\rangle)}2+2=\left\{\begin{array}{ll} x_{_T}+y_{_T}&\text{if } T \text{ is an axle}\\ x_{_T}+y_{_T}+1 &\text{if } T \text{ is a rim}\\ x_{_T}+y_{_T}-1&\text{if } T \text{ is generic or a spoke.}\end{array}\right.$$} Definition: For {$T\in \mathcal {B}^+$} {$$GCD_{_T}=\left\{\begin{array}{cl}(x_{_T},x_{_T})&\text{if } T \text{ is an axle}\\ (z_{_T},z_{_T})&\text{if } T \text{ is a rim}\\ (z_{_T},x_{_T})&\text{if } T \text{ is a spoke}\\ (x_{_T},z_{_T})&\text{if } T \text{ is generic.}\end{array}\right.$$} |