Re: [math-fun] some graphs for distributions modulo 1.
Simon writes: << I have been collecting some old ideas with graphs, especially the ones related to distributions mod 1, this is used to be my favorite hobby, . . .
. . . and then you hit puberty? Just kidding. These are fascinating graphics. Let me make sure I get it: Using various criteria, you get a whole bunch of empirical distributions mod 1, i.e. largish finite subsets of [0,1) -- and then you plot these on a circle by x -> exp(2pi i x), and finally you connect all pairs of these points on the circle to get a (largish) union of chords of the circle. (Or is something else going on?) Sometime in the past year I recall discussing (here?) the limiting *2D* distribution of [n points chosen from a circle at random, yielding all the points *inside* the circle determined by *intersections* of pairs of chords connecting the n points] -- as n -> oo. I think I see a general way to use the answer to this relatively simple question in order to describe the limiting 2D distributions suggesed by Simon's graphics . . . but more specificallly the limiting distributions of the *intersection points* of the chords (rather than those of the chords themselves, which looks a bit harder). --Dan
Yes, exactly that, each number in {0,1} is plotted on the unit circle with x in {0,1} -> exp(2*pi*i*x), and then from one x to the other a line is drawn. I find this way of representing numbers in {0,1} very practical since it gives a way to see if there are any patterns in the distribution. I am not saying that it is a way to detect any pattern, just that it finds many of them. I can explain some of them. simon plouffe
Quoting Simon Plouffe <simon.plouffe@gmail.com>:
each number in {0,1} is plotted on the unit circle with x in {0,1} -> exp(2*pi*i*x), and then from one x to the other a line is drawn.
This scheme works well in cellular automata theory. At least for one dimensional automata. 1) an automaton with cyclic boundary condition, connect a configuration to the evolved configuration. 2) de Bruijn diagram: connect neighborhood fragments according to represent the evolution of the complete neighborhood. In complex variable theory, illustrate Blaschke products (mapping the unit circle into itself) by connecting points to their images. - hvm ------------------------------------------------- www.correo.unam.mx UNAMonos Comunicándonos
participants (3)
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Daniel Asimov -
mcintosh@servidor.unam.mx -
Simon Plouffe