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The Mystery of the Sealed Box

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Teichmüller

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Štěpán

Notice that on the bottom the path moves SW-NE ↗ or NE-SW ↙ and on the top the path moves SE-NW ↖ or NW-SE ↘ .

The Two Corners Claim?: Seen from the top, as the path moves on the walls it always passes exactly two corners.

• The box is sealed because we don't care what's in it.
• {And in a sense it's not mysterious because all we're doing \\

is looking at straight lines on the surface of the box. {$45^\circ$}.}

The Mystery

of the


Sealed Box




Fred Henle and Jim Henle

This website is devoted to an investigation of a simple idea that leads to complex questions, some of which we can answer and some of which we can't (as yet). At this site we

• Tell you about the idea and the questions,
• Offer you programs to let you visualize and explore the questions,
• And the answers we have found,
• With proofs of our results, and
• With connections to many other problems and fields: Teichmüller Theory, translation surfaces, cutting sequences, episturmian sequences, symbolic dynamics, geodesics on polyhedra, and generalized continued fractions.

The box is sealed because we don't care what's in it. And in a sense it's not mysterious because all we're doing is looking at straight lines on the surface of the box.

And the lines are connected.

And they're always at an angle of 45° with the edges. The Mystery of the Sealed Box \\\\\\

The box is sealed because we don't care what's in it. And in a sense it's not mysterious because all we're doing is looking at straight lines on the surface of the box.

Sometimes the lines connect corners.

And sometimes the lines go on forever.

That's basically what we're looking at. Along the way we'll see

• A very pretty fractal
• A really surprising fact about the {$\pi\times \sqrt2\times e$} box.
• A new and wild definition of "greatest common divisor" for triples of numbers.

Here's some display math: {$$\sum_{k=0}^{n}n^2=\frac{n(n+1)(2n+1)}{6}$$}

Here's some inline math: {$e^{\pi i}+1=0$}

This is a test of inline GlowScript?:

⚠ (:glowbox 13 12 5:)

Did it work?