First Easy StepsBoxes with integral dimensions are compartively simple. Nothing Infinite: A geodesic passing through a point on a box with integral dimensions is either a loop or it connects in both directions to a corner. For boxes with irrational dimensions, complexities can occur. The following is the analytic version of the classification introduced in number theory. Definition: Let {$\mathcal C$} be the space of all unordered triples of real numbers (not necessarily distinct). Let {$\mathcal {C}^+$} be the set of unordered triples of positive reals. Think of {$\mathcal {C}^+$} as the collection of shapes of boxes. We will use "box" for a member of {$\mathcal{C}^+$} as well as for the geometric object. We will use angle brackets, {$\langle a,b,c\rangle$}, for members of {$\mathcal {C}$}. The Box {$\langle a,b,c\rangle$} is "triangular" if no member is greater than the sum of the other two. The box is "strictly triangular" if each member is less than the sum of the other two. The conclusions of What {$\Delta$} Does hold for {$T\in \mathcal {C}^+$} Definition: We say {$T\in \mathcal{C}^+$} is "normal" if for some {$k\in\mathbb N$}, {$\Delta^k(T)$} is strictly triangular. A "rim" is a box {$T\in \mathcal{C}^+$} such that for some {$k\in\mathbb N$}, {$\Delta^k(T)$} is triangular but not strictly triangular. If a box is not a rim and not normal, we will say it's "abnormal". There are abnormal boxes: There exist boxes {$T$} such no {$\Delta^k T$} is triangular. It's not normal to be loopless: Every normal box has a loop. Some rims have loops, some don't: The rim {$a\times b\times c\ $} has a loop iff {$\ \frac ab\ $} is rational. The union of rims and abnormal boxes is the Rauzy gasket. |