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The Least Common Multiple of a Box

For rectangles, the length of the path of the billiard ball from one corner is the least common multiple of the dimensions of the rectangle. For Boxes there can be paths from corners of different lengths.

At Most Three Paths: For {$T\in\mathcal {B}^+$}, there are at most three different path lengths (from corner to corner). If {$T$} is not an axle, there are at most two different path lengths.

Definition: For {$T\in\mathcal {B}^+$}, let {$L^1_{_T},L^2_{_T},L^3_{_T}$} be the path lengths, where

• if there is only one path length, {$L^1_{_T}=L^2_{_T}=L^3_{_T}$},
• if one path length has eight specimens, that length is {$L^1_{_T}=L^2_{_T}$}
• if there are three different path lengths, the order is arbitrary.

Let LCM{$_{_T}\in\mathbb {R}^2$} be the vector, {$(L^1_{_T},L^2_{_T}+L^3_{_T})$}.

Note that the total, {$L^1_{_T}+L^2_{_T}+L^3_{_T}$}, is one-fourth of the sum of all twelve paths.