Some rims have loops, some don't:The rim {$a\times b\times c\ $} has a loop iff {$\frac ab$} is rational. Proof: By Loopless, it is sufficient to consider boxes that are triangular but not strictly triangular (note that if one ratio of dimensions is rational, all three are). Rational Claim: For the rim {$a\times b\times c\ $}, {$\frac ab$} is rational iff {$\frac ac$} and {$\frac bc$} are rational. Consider, now, the triangular box {$a\times b\times (a+b)$}. We'll imagine part of a geodesic on the box and take the overhead view.  The geodesic moves around the rectangle in steps, a diagonal step on the bottom or top, a move around the outside (climbing up or down) then a diagonal step, then an outside step, and so on. Clearly, the distance between the point between steps and the corners on that side is a linear combination of {$a$}, {$b$}, and {$x$}.  Division Claim: On the top edge and bottom edge of the rectangle (which is the top/bottom of the box) the distance to the corner ahead (as the geodesic moves around the box) is of the form {$mb-na-x$}, where {$n$} and {$m$} are natural numbers. On the sides of the rectangle, the distance to the corner ahead is of the form {$na+x-mb$}. Furthermore, with each step the coefficients do not decrease and with each diagonal step one coefficient increases and the other stays the same. The Claim shows that if there is a loop, then {$\frac ab$} is rational. In order to have a loop, we would have return to where we started. But then we have {$$x=na+x-mb,$$} so {$$\frac ab=\frac mn.$$} For the other direction, if {$\frac ab=\frac mn,$} {$m$} and {$n$} integers, then consider the {$2na\times 2nb\times 2n(a+b)$} box. It has the same shape as the {$a\times b\times (a+b)$} box, it's sides are integers, moreover, they are even integers. By Magnified Boxes Have Loops this box has a loop. So the original box has a loop. |