Classification of Boxes We use {$a_{_T},b_{_T},c_{_T}$} and {$x_{_T},y_{_T},z_{_T}$}, as defined in A Euclidean Algorithm for Boxes. Definition: A box {$T\in\mathcal {B}^+$} is an "axle" if {$z_{_T}=0$}. Equivalently, an axle is a box where {$x_{_T}=y_{_T}$}. Equivalently, an axle is a box where {$a_{_T}=b_{_T}=c_{_T}=x_{_T}$}. In the box triangle, axles are the centers of the triangles. Definition: A box {$T\in\mathcal {B}^+$} is a "spoke" if it is not an axle, but the two largest of {$a_{_T},b_{_T},c_{_T}$} are equal. In the box triangle, spokes connect the centers of the triangles with their vertices. Definition: A box {$T\in\mathcal {B}^+$} is a "rim" if {$x_{_T}=0$}. A triangular rim is a box of the form, {$a\times b\times (a+b)$}. In the box triangle, rims form the borders of the triangles. If we think of a rectangle as a box of height 0, then rectangles are rims. Definition: Boxes which are not axles, spokes or rims are "generic". |