The Box TriangleThe Box Triangle is a way to organize the universe of box shapes. Every box has three dimensions, so the universe of box shapes would seem to be three-dimensional. But if our interest is in where the path from a corner leads, then we can shed one dimension. The reason is that the scale of the box is irrelevant. That is, for example, the picture for the {$3\times 5\times6$} box is the same as the picture for the {$9\times 15\times18$} box.  So if we want to work with all possible box shapes, we can simply look at all boxes where the dimensions add up to 1. A nice way to visualize all triples, {$a,b,c$} with {$a+b+c=1$}, is to look at all points inside an equilateral triangle with height 1.  For any point in the triangle, the sum of the distances to the sides  adds to 1. (The area of the triangle, {$\frac s2$}, is equal to the areas of the three smaller triangles, {$\frac {as}2+\frac {bs}2+\frac {cs}2$}.)  Thus, boxes of every shape correspond to points in the triangle and vice-versa. We'll use ordered triples, {$(a,b,c)$}, to refer to points in the triangle. The distances {$a$}, {$b$}, and {$c$} are called the "barycentric coordinates" of the point. |