Questions We Still Have:

On integral boxes:

  1. We still don't have a closed-form expression for the length of the path from the corner. Perhaps this is not possible.
  2. Is it possible to tell if the {$a\times b\times c$} box is generic without repeatedly applying {$\Delta$}?
  3. Is there an operation for which the GCD is really the greatest common ... ?
  4. Is there an operation for which the LCM is really the least common ... ?
  5. For what boxes is there more than one different loop length?
  6. How many possible loop lengths can there be on a box?
  7. Is there anything one can say about how destinations or loops behave under the {$C$} mapping ([A Generalization of Continued Fractions]])?

On analytic boxes:

  1. Do abnormal boxes have loops?
  2. Which boxes have paths infinite in both directions?
  3. Is there a nice geometric description of the action of {$\Delta$} on the box triangle.
  4. What can be said about boxes with one or more negative dimensions?
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Page last modified on June 25, 2013, at 07:15 AM