Hi all, Lately, I've been playing with the signature of irrational numbers. My understanding, from what I've read on the web, is this: For a positive irrational number x, form the numbers y = i + j*x, where i and j are both positive integers. Since x is irrational, no 2 y values will be the same for different i and j. Arrange the ys by size, then the sequence of i values is a fractal sequence, and the signature of x. For x = phi ~ 1.618034, the first few entries are: i j y 1 1 2.618033989 2 1 3.618033989 1 2 4.236067977 3 1 4.618033989 2 2 5.236067977 4 1 5.618033989 1 3 5.854101966 3 2 6.236067977 5 1 6.618033989 And the signature begins: 1, 2, 1, 3, 2, 4, 1, 3, 5. If you strike the first occurence of every integer in the sequence, you get the original sequence back, which makes this a fractal sequence. My questions are: - What about the j sequence? From what I've seen experimentally, it seems to be a fractal sequence, too. Why is the signature the i sequence as opposed to the j sequence? What's known about the relation of the j sequence to the i sequence? - For a limited set of integers (both i and j run from 1 to 50), I plotted i vs. j, and the result was very interesting (I thought). You can find the picture here: http://www.fractalus.com/kerry/sigofphi.html The plot is one continuous zig-zag line which seems to never cross itself. But, the angle of the line changes slightly, causing some areas to bunch up and appear darker, and others to spread out and appear lighter. The overall effect is of a series of rectangles drawn in different shades of gray. Can anyone point me to other work that has been done on this? Thanks, Kerry Mitchell -- lkmitch@att.net www.fractalus.com/kerry