I have nothing substantive to say, but I thought others might enjoy an example random diagonals, which is drawn from an earlier MO question, "[Shortest grid-graph paths with random diagonal shortcuts][1]": <br /> ![Random Diagonals 50x50][2] <br /> [1]: http://mathoverflow.net/questions/45920/ [2]: http://cs.smith.edu/~orourke/MathOverflow/RandomDiagonals50.jpg