Window
Size (iterations). [BLUE
HIGHLIGHTS].
Dynamics of Recursive Differences of
Differences
When basin length is NOT a power of 2:
for a basin with L=5
Hierarchies formed when basin length was a power of 2. On the webpage that sent you here, in the previous section considered how hierarchies form when differences are taken recursively in the differences in basin dynamics when basin length was a power of 2.
What happens when L not equal to a power of 2? In this introductory example, the applet allows you to observe for yourself what patterns form across TAO levels.
DRAFT: REVISE TEXT
Intructions for using the the applet
Using the applet.
Set "Window Size (Iterations)" to 7 or 8
Check "Use Delay" and use the Slider to set the delay between iterations to about 10 ms. What works depends on your particular computer's speed and video card. If the your computer runs the pattern too fast, make the delay longer. If the your computer runs the pattern too slowly, uncheck "Use Delay." We recomend adjusting the delay so that the speed (next to the Play controls reads around 20 to 30 iteratins per second (ips) [see instructions].
Press Play (green arrow).
Standing Waves: Window Size = a factor of 5. With Window Size set to 5 or 10 you should see (as in other examples) the flow stopping to form standing waves that appear like static patterns. As in other cases, they are not static; they are simply in a phase relationship with the how we are representing the system's dynamics. You can confirm that "static" standing waves" are actually dynamic by watching the ips speed indicator while you either change window size or change the delay setting. The ips will change value even when the basin and TAO patterns appear static because they are actually the output of process. They require the process to run in order to exist. As we have said, this is a fundamental realization of considering the universe to be purely dynamic relations: That which appears to be things are actually sets of dynamic relations.
TAO levels perceived from Standing Waves. Look at the "static" patterns across TAO levels. Several observations stand out. Unlike the case when L equal a power of 2, the derivatives do not go to zero within basin length. In fact, they don't go to zero at all.
META BASINS: Derivatives repeat themselves. If you do the careful counting (or eyeballing) youwill observe that TAO-1 repeats itself exactly at TAO 16 (for this specific case). The output in this example does not allow you to observe this directly, but the general pattern is that TAO-1 will repeat exactly at TAO-16 and again at TAO-31 and again at TAO-46, and so on. Similarly, TAO-2 will repeat at TAO 17, 32, 47, ... And TAO-3 will repeat at TAO 18, 33, 48,... The derivatives of the dynamic differences in the original basin, applied recursively, themselves fall into a basin.
From at static analysis, we at first thought that there were 15 meta basins each of meta-length, ML=15. That is in some sense descriptively accurate, but it misses the the most penetrating insight available in this context. To set up that insight, we must leave the staticness of standing waves and examine representations where the dynamics are more obvious.
Set Window Size to 6 or 7. You may have to adjust the Delay between interations to get the output speed to between 20 and 30 ips, which is ideal for perceiving the patterns we will next discuss.
Three Meta-Pattrens in the flow of TAO. You will observe what appears to be three distinct patterns flowing within the TAO levels. The repetition of pattern is not 15 long, it is only three. What are the implications of these meta-patterns in the TAO levels?
There is much to say about this interesting behavior of differences in differences.
Instructions Top
Window
Size (iterations). [BLUE
HIGHLIGHTS].
This is the most important control conceptually because it adjusts the phase relations between basin length and the processing of the basin (and this allows the extraction of different aspects of the dynamic patterns portrayed here.
Drag the Slider (highlighted in blue) from its default setting of 60 down to whatever size is suggested above for using the particular applet you are viewing. In this case the slider has been dragged down to 11 iterations.
Click the Slider Bar. Optionally you may click on the Slider Bar and the Window Size will slowly scroll in the direction (to right of slider or to the left of slider) you want. You can easily get a change of one unit by clicking on the slider bar. This gives you a finer degree of control over window size than does dragging the slider.
Read the Window Size. To the right of the Window Size area is a number (highlighted in blue) that tells you exactly what the Window Size is. Top
Setting Delay. [YELLOW HIGHLIGHTS].
WHY?: Adjusting your Computer's speed to your Monitor's speed: Most monitors cannot paint accurately faster than 66 to 77 times per second. In this class dynamic systems, we ask the computer to paint each iteration of the system to the screen. Depending on the how fast your computer is (it's clock speed mega-Hz or giga-Hz and what type of video card it has) this software may send requests to paint images 1000 or more times per second. Once the iterations per second is higher that 65 or 70 iterations per second (ips) what you see on the screen is some undetermined interaction between your monitor hardware and the behavior of the dynamic system. In other words, you aren't seeing the behavior of the system any more, you are seeing that part of the behavior that the screen happens to capture.
Solution. Click on the Use Speed radio button (higlighted in yellow). Then drag the Delay (between iterations) Slider (highlighted in yellow) from its (very slow) 250 millesecond delay between iterations down to some lower value that gives you a good sense of dymanic motion in the output. As we said, you generally want the the iterations per second (ips) to between 20 ips and 60 ips, although it seems to work well in this case as low as 6 to 8 ips. You your judgment as this is about your perception.
If your computer is slow, you may not need to use the Speed control. Top
Interations per Second (ips). [HIGHLIGHTED IN LAVENDER].
When you push PLAY, you generally want to have the iterations per second indicator (just to the left of the double black arrow on the control bar) to be between 20 and 60 ips. This range allows you to perceive apparent motion effects but is within your monitor's ability to paint the screen. Obtaining this range may require setting the delay (see below) between iterations.
Sizing the Viewing Area. [ORANGE HIGHLIGHTS].
Resize Viewing Area. [ORANGE HIGHLIGHTS]. The dimpled bar between the Controls and the Output Frmae (Viewing Area) can be dragged in either direction (as indicated by black arrows). This allows you to adjust the viewing area to see more or less TAO levels.
Full-Interface Viewing Area. [ORANGE HIGHLIGHTS (Top)]. Once you have the controls set as you like them you can eliminate them and see more derivatives by clicking on the little left-pointing arrow at the top of the bar dividing the controls from the output frame. Clicking the right-point arrow will return the controls to veiw. You can also grab the dividing bar and drag one way are another to size the parts of the interface the way you want.
Viewing the Iterations per Second (ips) indicator. To see the number indicating the iterations per second (ips), it may be necessary to drag the dimpled bar to the right. This is usually not the case, but when it is true it is minor and easy to do but it is an annoying adustment that makes these instructions appear to more complicated than necessary. We are working on fixing that.