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Boxes with Negative Height

We take the overhead view to look at boxes {$a\times b\times c$} with {$c<0<a,b$}.

We declare that when a path hits an edge of either the bottom or the top, it reverses course. The best way to see that is to look at a typical path.

Now intead of winding around clockwise, the geodesic goes counter-clockwise, a distance of {$c$}.

and now it reverses again.

This is actually the start of a loop. In its entirety, the loop looks like this:

And in three dimensions, it looks like this:

The {$3\times 4\times 5$} box, for example, has 8 loops. Surprisingly, the {$3\times 4\times -2$} box has exactly 8 loops too, with the above description of geodesic.