Summary Chart of Correspondences Between Rectangles and Boxes

 The {$m\times n$} rectangle {$R$} The generic, triangular {$a\times b\times c\$} box {$T$} Euclidean algorithm subtract smaller from larger subtract smaller two from largest Number(s) from Euclidean algorithm {$(m,n)$} {$a+b-c,(c-b,c-a)$} GCD GCD{$_{_R}=(m,n)$} GCD{$_{_T}=(a+b-c,(c-b,c-a))$} Is loopless GCD{$_{_R}<2$} {$||$}GCD{$_{_T}||<2$} Number of Loops GCD{$_{_R}-1$} GCD{$_{_T}\cdot(4,2)-6$} Paths from corners always end always end Destination of path from corner Depends on evenness of {$m,n$} Depends on evenness of {$c-a,c-b$} Path from corner is infinite iff {$\frac nm$} is irrational iff {$\frac {c-b}{c-a}$} is irrational Length of Path from corner {${[m,n]}$} {$L_1,L_2,L_3$} LCM LCM{$_{_R}=[m,n]$} LCM{$_{_T}=(L_1,L_2+L_3)$} Product {$m\cdot n$} {$a\cdot b+b\cdot c+c\cdot a$} Big Theorem GCD{$_{_R}\cdot\$}LCM{$_{_R}=\$}Product GCD{$_{_T}\cdot\$}LCM{$_{_T}=\$}Product Continued something or other {$(m,n)\mapsto\left\{\begin{array}{l}(m+n,n)\\(m,m+n)\end{array}\right.$} {$(a,b,c)\mapsto\left\{\begin{array}{l}(a+b+c,b,c)\\(a,a+b+c,c)\\(a,b,a+b+c)\\(b+c,a+c,a+b)\end{array}\right.$} rationals, rational pairs {$(m,n)\Longleftrightarrow\ \frac mn$} {$(a,b,c)\Longleftrightarrow\ (\frac a{a+b+c},\frac b{a+b+c})$}