| | 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})$} |
