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Summary Chart of Correspondences Between Rectangles and Boxes

  The {$m\times n$} rectangle {$R$}The generic, triangular {$a\times b\times c\ $} box {$T$}
Euclidean algorithmsubtract smaller from largersubtract smaller two from largest
Number(s) from Euclidean algorithm{$(m,n)$}{$a+b-c,(c-b,c-a)$}
GCDGCD{$_{_R}=(m,n)$}GCD{$_{_T}=(a+b-c,(c-b,c-a))$}
Is looplessGCD{$_{_R}<2$}{$||$}GCD{$_{_T}||<2$}
Number of LoopsGCD{$_{_R}-1$}GCD{$_{_T}\cdot(4,2)-6$}
Paths from cornersalways endalways end
Destination of path from cornerDepends on evenness of {$m,n$}Depends on evenness of {$c-a,c-b$}
Path from corner is infiniteiff {$\frac nm$} is irrationaliff {$\frac {c-b}{c-a}$} is irrational
Length of Path from corner{${[m,n]}$}{$L_1,L_2,L_3$}
LCMLCM{$_{_R}=[m,n]$}LCM{$_{_T}=(L_1,L_2+L_3)$}
Product{$m\cdot n$}{$a\cdot b+b\cdot c+c\cdot a$}
Big TheoremGCD{$_{_R}\cdot\ $}LCM{$_{_R}=\ $}ProductGCD{$_{_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})$}
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Page last modified on January 01, 2014, at 03:03 AM