Proof of concept. Made with Processing.
Four closed loops, one open string from the center of them. On these strings information is stored: white or black. When the information crosses other information, they interact and change color. Depending on the rules underlying these interactions, and the arrangements of the strings, patterns emerge.
When the string crosses the first loop from the inside, it will always be white. The string and the first loop change color, and when it loops back into itself it changes again: from white to black to white again.
The second loop is crossed by a string that periodically changes colors, but that period doesn't sync up to it's own period. Therefore, the pattern increases in complexity. In some ways, it's similar to the superposition of two waves forming a beat. As the loop feeds back into itself, the beat "echoes", changing the pattern every full cycle of the second loop. If one would wait for long enough (longer than this animation) the "echoes" would cancel each other out and there would be a moment where the two innermost loops would be enterely black. And if you waited equally long after that, they both would be white again.
The third and fourth loops increase complexity of the pattern yet again (if any pattern is left). Maybe, if the animation would run for long enough, there would be a moment where all four loops are black (and then after wainting the same amount of time white again), but I don't know that for certain. It would take an awful long time though.
No, I haven't read GEB or I Am A Strange Loop yet. This work was actually inspired by Jeff Hawkin's talk on brains at TED: