Ooh, a fractal thread! I finally have an excuse to show off cool stuff I learned in Chaos Theory last quarter.
First cool thing: Look at the binary expansions of the points on [0,1]x[0,1]. If the ordered pair is of the form (0.1a2a3a4… ,0.1b2b3b4… ) leave that point white; same for (0.a11a3a4… ,0.b11b3b4… ), and so on. If both numbers have a 1 in the same place anywhere in their binary expansions, leave it white, otherwise paint it black. An obvious consequence is that there can’t be any black points in the upper-right quadrant square, or the upper-right subquadrant of the quandrants, and so on.
If you want a less mathy idea of what just happened, think of quartering a square, removing the top right piece, quartering the three leftovers, and removing the top right square from them. Repeat this procedure an infinite number of times, and you’ll end up with the same collection of points. Now here’s the crazy part: what you get is just the Sierpinski gasket, but made out of right triangles.
Well yeah, you got the ide- . . . i see wat u did there
Okay, if the sequence of ones in your example terminates, then we’re cool. That’s kinda why they call [0,0.1) a half-open interval.
well okay maybe not why, but you know . . . I liked it better when I was the pedant.