10/21/2007, 05:35 AM

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.

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