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Generic structure for slog
#1
Okay, to help us better understand the structure of the slog (base e), I think it's important to look at it generically. In other words, let's be ignorant for a moment about the exact values of the slog, and explore instead the branches and their layout relative to each other and relative to fixed points (or singularities, as I'll also typically call them).

To do this, let's start with the following region and call it the "(generic) critical region". The borders of this region are, on the left, the straight line drawn between the two primary fixed points (0.318+1.337i and its conjugate), and on the right, the exponentiation of this line, an arc with center at the origin and endpoints at the fixed points:

   

Note that this is a fairly generic region, but it suffices for our purposes. No point in the region can be exponentiated without ending up outside the region, and if we take the logarithm of any point in the region, we again end up outside the region. Exponentiate any point on the left boundary, and we'll get a point on the right boundary. Conversely, take the logarithm of any point on the right boundary, and we'll end up with a point on the left boundary. And of course, the fixed points complete the enclosure of this region.

I've color coded this region in a fairly smooth manner, such that green corresponds to the left side of this interval, and blue to the right side. The exact method I used to determine this "smooth" coloring is unimportant, because as I said, this is supposed to be generic.

Now, let's logarithmicize this region. Yes, I made that word up. We end up with the following:

   

Notice that copies of the new region at the origin appear at 2*pi*i intervals. In fact, though not seen in this relatively small image, there are an infinite number of copies. These correspond to the various branches of the natural logarithm. Exponentiate any one of these regions, or indeed all of them together, and we get back the critical region (or multiple copies of it).

The second thing to notice is that, if we are to maintain some degree of continuity, it only makes sense to make copies of the critical region at 2*pi*i intervals:

   

And, of course, if these other regions are green-blue (just like our critical region), it only makes sense that a red-green region should appear when we logarithmicize these new regions:

   

Let's zoom in to try to make out a little more detail:

   

Still not much to see. Don't worry, it'll start to fit together soon. And by the way, future zooms will not be centered at the origin, for reasons that will be obvious.

You may at this point have doubts that this duplicating of regions is justifiable, but as we go further, you'll see the regions connect smoothly. The "smoothly" part would indicate (qualitatively) that the graph is continuous, and differentiable at least once. I've also given a more mathematical justification elsewhere, but here, we're mainly concerned about getting a quantitative feel for the slog.
~ Jay Daniel Fox
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#2
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).
~ Jay Daniel Fox
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#3
Okay, before we logarithmicize again, let's zoom in one more time. Please note that the tick marks now correspond to 1/4 unit spacings, with larger ticks at the units.

   

The first thing to notice is that, at this zoom level, it becomes clear that those "random dots" I mentioned previously are now properly connected, though new "random dots" have appeared. As we continue to fill in the graph, these will of course continue to connect up with the main graph.

Also, please note that the green at the boundary of the critical region is NOT the same green as the branches off the ribs. My color palette cycles every three iterations, so for now we have to pay attention. By the time I do graphs with the Andrew's slog, I'll have a color system worked out. For now, I'm rather limited, since I'm using a hack in SAGE to generate these graphs. (For the curious, I'm using a matrix_plot with a 512x512 matrix, and the image size just right to get one pixel per element (6.13x6.13), and I'm using the "hsv" color map with an algorithm that jumbles the colors enough to get 1024 distinguishable colored intervals out of a 256-color palette.)

Okay, now, another iteration:

   

Notice that the detail gets too fine to make out. However, what should concern us isn't that the detail is so fine, but that to get from one green region to another, we have to pass through the magenta. And by analogy, between each magenta spoke is another, even more densely packed region, and so on to infinity, such that we can't actually cut a path through. We must go back up to the bluish rib to get from one green branch to another. Likewise, we can't get from one rib to another by going straight down. We must backtrack, back up to the backback.

Okay, I'm out of pictures for now. Time to make more. I'll try to show an exponential branch at some point as well, but for now, I want to go one iteration further into the logarithmic branches.
~ Jay Daniel Fox
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