Almost Claim:Almost every box's dimensions are linearly independent over the rationals. Proof: We're using Lebesgue measure on the box triangle. If the dimensions, {$a$}, {$b$}, {$c$}, are dependent, then for some rationals, {$p$}, {$q$}, {$r$}, {$$pa+qb+rc=0.$$} But {$c=1-a-b$} so we really have {$$r=(r-p)a+(r-q)b.$$} In other words, the box, as a point in the box triangle,  lies on a line connecting two rational points,  that is, points with rational barycentric coordinates.  There are only countably many such lines and each can be (inside the box triangle) enclosed in an open set of arbitrarily small measure. Hence, the set of linearly dependent triples has measure 0. |