The Mystery

of the

Sealed Box

Fred Henle and Jim Henle

This website is devoted to an investigation of one aspect of plain, ordinary boxes. The aspect is utterly simple but it leads to issues of great complexity. All we will be doing is looking at straight-line paths on the surface of the box, that is, geodesics. We have many questions, and we have some answers.

By way of introduction, we offer---

- An overview of (sealed) boxes.
- The motivating context of billiards on rectangles.
- Graphics programs? which visitors can use to visualize and explore.

We organize our work under these general headings:

And in summary:

- Summary Chart of Correspondences between Rectangles and Boxes
- Questions we still have
- Connections to other fields: translation surfaces, square-tiled surfaces, Teichmueller theory, cutting sequences, episturmian sequences, symbolic dynamics, geodesics on polyhedra, and generalized continued fractions.
- People we want to thank.

Highlights include:

- A very pretty fractal

- A really surprising fact about the {$\pi\times \sqrt2\times e$} box.

- A bizarre sort of number theory where 3, 4, and 6 are "relatively prime"---but 3, 4, and 5 are not.
- An odd use of the Rubik's cube to compute trajectories.

- A map, the three-coloring of which is equivalent to following a geodesic around the box.

Retrieved from http://enigmatists.net/Boxes/Boxes

Page last modified on August 28, 2013, at 11:32 PM