10/21/2007, 04:54 AM
Now, let's logarithmicize again:
First of all, I should explain the little random-looking dots that just appeared. Those are the free-floating red-green regions from the previous graph, logarithmicized. Don't worry about them for now, if they're bothering you.
Notice that we've nearly filled in the entire region to the left of the line with real part 0.31813. This is what I've come to call the "backbone" of the slog. Why I call it this becomes even more clear if we logarithmicize once more:
And there you have it. Alternating projections, like ribs coming off a backbone. Between each of the vertebrae is a disk, or in this case, the critical region and its images at 2*pi*i intervals. And notice that it all fits together smootly, as promised.
Notice that I tried to set up the coloring scheme so that detail is exposed in this region. You'll need a true-color display to see it, but it's worth taking a close look. Even though this isn't the real slog, we may as well try to expose subtle details. In fact, let's zoom in:
BTW, each of these new projections (the "ribs") will stretch out to a positive infinity (i.e., with real part going to positive infinity). And before we do so, notice that if we try to logarithmicize one more time, we'll have overlap with the critical region. This is due to the fractal branching of the slog. We can only wrap roughly four "unit" regions around a singularity before we overlap. The best we can do for now is to remove one iterated region from the graph as we add another. In this case, we'll take one region off the "exponential" end of the graph, to make room for a logarithmicized one. (We could go the other way, of course, to analyze the exponential branches, which I'll do in a later post.)
The fun thing to note about the new green regions here (the "ribs" after being logarithmicized), is that they group so densely that we start getting funny-looking patterns on the right side of the image. This is due to aliasing problems. The detail is simply to fine at the resolution we're using. We'd need to zoom way in to be able to make out any useful detail over there.
Notice that I've been zooming off the real line. To keep things connected, I focussed on the primary upper logarithmic "rib" off the backbone. As we desire to zoom in further in subsequent iterations, we'll actually focus on the lower half of this particular rib (i.e., branch).
First of all, I should explain the little random-looking dots that just appeared. Those are the free-floating red-green regions from the previous graph, logarithmicized. Don't worry about them for now, if they're bothering you.
Notice that we've nearly filled in the entire region to the left of the line with real part 0.31813. This is what I've come to call the "backbone" of the slog. Why I call it this becomes even more clear if we logarithmicize once more:
And there you have it. Alternating projections, like ribs coming off a backbone. Between each of the vertebrae is a disk, or in this case, the critical region and its images at 2*pi*i intervals. And notice that it all fits together smootly, as promised.
Notice that I tried to set up the coloring scheme so that detail is exposed in this region. You'll need a true-color display to see it, but it's worth taking a close look. Even though this isn't the real slog, we may as well try to expose subtle details. In fact, let's zoom in:
BTW, each of these new projections (the "ribs") will stretch out to a positive infinity (i.e., with real part going to positive infinity). And before we do so, notice that if we try to logarithmicize one more time, we'll have overlap with the critical region. This is due to the fractal branching of the slog. We can only wrap roughly four "unit" regions around a singularity before we overlap. The best we can do for now is to remove one iterated region from the graph as we add another. In this case, we'll take one region off the "exponential" end of the graph, to make room for a logarithmicized one. (We could go the other way, of course, to analyze the exponential branches, which I'll do in a later post.)
The fun thing to note about the new green regions here (the "ribs" after being logarithmicized), is that they group so densely that we start getting funny-looking patterns on the right side of the image. This is due to aliasing problems. The detail is simply to fine at the resolution we're using. We'd need to zoom way in to be able to make out any useful detail over there.
Notice that I've been zooming off the real line. To keep things connected, I focussed on the primary upper logarithmic "rib" off the backbone. As we desire to zoom in further in subsequent iterations, we'll actually focus on the lower half of this particular rib (i.e., branch).
~ Jay Daniel Fox
