I ran across this years ago. It’s defined by the addition rule: a/b + c/d = (a+c)/(b+d) (Sounds crazy – until you think of x/y as meaning There’s a pile of x things on one side, and one of y things on the other.) Applying it in almost a Pascal's Triangle sort of way generates ... er, a triangle (see below).. ..which I think has the properties of the sequence: 1/2, 1/3, 2/3, 1/4, 3/4, ... (as you see, skipping non-reduced fractions) Properties: One person put it this way: "In 1873 Georg Cantor used an argument similar to this to show the existence, not of irrational numbers within every interval, but of transcendental numbers every interval." Start with 0/1 on the left, and 1/1 on the right of the top line. To form each item of the next line, rational-add the two fractions - anywhere above - that are closest on either side to its position in the line. (Hard to articulate: See the first four lines that of the result from that, below.) (I'm using dashes as separators since so many editors like this one collapse strings of blanks into one.) 0/1---------------1/1 ---------1/2 ----1/3-------2/3 -1/4-2/5--3/5-3/4 I'm looking to references, but of course first-hand knowledge is great.