There are abnormal boxes:There exist boxes {$T$} such no {$\Delta^k T$} is triangular. Proof: We seek a box, {$T=\langle a, b, c\rangle$} with {$a< b<c$}, {$a+b+c=1$}, such that applying {$\Delta$} to {$T$} gives us {$T$} back again, that is, when we normalize {$$\langle a, b, c-a-b\rangle,$$} we get {$$\langle \frac ac, \frac bc, 1-\frac ac - \frac bc\rangle$$} and {$a=1-\frac ac - \frac bc$}, {$b=\frac ac$}, and {$c=\frac bc$}. These equations together with {$a+b+c=1$} give us {$$b=c^2 \text{ and }\ a=c^3$$} and eventually, {$$ c^3+c^2+c-1=0.$$} It's not hard to show that this equation has a single root, {$c\approx .5437$}. It follows that {$\langle c^3, c^2, c\rangle$} is abnormal. There are many more. |