Here is the traditional picture, from Propp's site. I have photographed it so now it is a jpg in which individual pixels of the picture are not discernable, so don't strain yourself trying to see details of the picture. Print out your own picture from the link above if you want to see the details! Then you can have the fun of drawing your own lines, too.

You can see I have drawn a bunch of lines on it. After trying many pens and pencils in vain, I found that a Sharpie permanent marker will satisfactorily write on top of the laser printer inks.

And here is the directed graph picture (where each rotor routes in just two opposite directions, and N/S rotors are interleaved with E/W rotors in a checkerboard pattern).

Again, I have drawn curves that meet in the splotches of the picture. A splotch is a place that is locally a quilt of fair-sized uniform parts. You can see some splotch spots on the right hand side, where they are not obscured by my lines.

The red lines visible in the upper left correspond to the lines in the traditional picture. Here's a close-up:

You may be wondering what the circles and X's are. The circles are where two red lines meet. The X's are on circles that shouldn't be there. (When you're using a Sharpie permanent marker, you can't erase!)

You can compare the red lines with a close-up of the traditional picture:

You can see how the blotch on the circumference at 45 degrees is the center of an extremely warped square!

Also note that if you look at the "blue corner" and the "orange corner" of each square, the pattern is periodic in the same way that the squares are. This suggests avenues for further exploration.

I also looked at how the lines of the "directed" picture (or at least a subset of them) fit onto the traditional picture:

You can compare it with part of the "directed" picture, and see that the "directed" picture has twice as many lines:

Of course, you can make better pictures than these by using your own color printer, and perhaps even having the computer draw the lines. (But it's more fun to draw them yourself.)

To have the computer do it, just transform the square lattice via the complex plane mapping that maps z into 1/sqrt(z). (Transformation discovered independently by me and Dan Hoey, him first.)

That's all folks!

page created on 09-24-2005