Coloring maps to plot geodesics

Let's say you are looking at the {$3\times 9\times 4$} box.

Start with one quadrant of the plane and a line at 45 degrees.

You are going to create a map in the quadrant using only infinite rectangular strips of three widths (3, 9, 4) and three colors. Since the base of the box is {$3\times 9$}, we start with these. First a strip of width 3.

then a strip of width 9.

Note the black dot. It's above the 45 degree line, so the next strip must be vertical. It will be next to the blue and green strips, so it must be a third color. It's a strip of width 4.

The black dot is still left of the line. But now the strip must be blue.

Now the dot is to the right of the line, so we want a horizontal strip. Coloring demands that it's yellow.

We just keep going this way ...

... until finally the black dot is on the red line.

This corresponds to the geodesic:

We haven't explored this. It might be useful sometime.

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Page last modified on May 24, 2013, at 01:30 PM