Magnified Boxes Have loops:For any {$\langle a,b,c\rangle$}, {$k>1$}, the box with dimensions {$ak\times bk\times ck$} has a loop. Proof: Locate the box in the first octant with one corner at the origin (standard position). Imagine a point, {$(x,y,z)$} moving along a path that begins at the origin. Let {$g(x,y,z)$} be the number of coordinates which are {$\equiv 0 \mod k$}. At the start, {$g(0,0,0)=3$}. At the next step we have {$g(1,1,0)=1$}. When the path reaches the first edge, we have either {$g(ak,ak,0)=3$} or {$g(bk,bk,0)=3$}. At every edge we will have, {$g(x,y,z)=3$}. Consequently, a paths that begin at a corners will never pass through the point {$(1,0,0)$}, since {$g(1,0,0)=2$}. Thus, there must be a loop. |