The BIG problem with adding more parameters to an animation is the number of events grows exponentially. My forte with one parameter animations often uses a thousand quaternion events. A million quaternions would be needed for two parameters, yikes. My computer often gags on that many quaternions (other people have a much lower tolerance, an integer less than one). Go up to a 4-parameter animation, and one would need a quadrillion events. Even an animation that needs 100 steps for each parameter means a hundred million events to plot. The software was written to put a limit on the number of events.
Start simple. I have as many as four parameters, a, b, c, and d. Here is my arbitrary map between the parameters and the plotted values of t, x, y, and z.
If the program is given a starting parameter of -3 and and end of 4, then only the first parameter a is used.
Some consider this one parameter animation quite dull. What I find exciting is that the simplest operation in the quaternion world - repetitive addition - creates an inertial observer, a deep idea in special relativity. I learned way back in second grade that multiplication is repetitive addition. It is interesting to me that squaring a quaternion generates the flat spacetime interval. There are others who do not find that observation at all interesting, in fact, downright misleading. Some care was used to find the minimum number of events to make a smooth animation, about 40.
Step up to the two parameter world by feeding the program an array of start points, "-3, 2" and end points "4, -2":
This results in a string that moves around spacetime. The animation is over with quickly, going through 1600 events.
The third step needs three numbers to start, and three to end.
The result is a membrane that meanders by churning through 64,000 events.
So the next step up should be easy enough: just add a fourth parameter, and one should see a solid. I did the exercise, and the animation looked...just like the last one.
I spent much time debugging: where did I go wrong? The more basic question becomes how does one debug? There is the age old practice of staring at the code, trying to see if there is a misplaced number or variable. I was not able to spot it, the code is pretty simple written in Perl. Maybe it was an issue involving the function or input values. I did an experiment on two variations for the mappings between the parameters and the values in spacetime. One covered all the bases, the other was an isolationist:
All the Bases:
It seemed like the isolationist had to make a volume of points, something more than a membrane, which it did. It formed a solid block that sat there, never budging. The All the Bases looked exactly like the one parameter expression. While the starting and ending points might be different for the four parameters, they all contribute to the spacetime quaternion in exactly the same way. The map was too symmetric. I decided to try a variation that would break the symmetry in a simple way:
All the Bases Broken:
This does generate a cube that moves in spacetime:
Nice. It looks like some sort of magic trick, making a cube out of nothing that then disappears. My initial arbitrary map had a symmetry made clear at first by the animation.
Simple addition follows a simple pattern up the ladder of permutations: a dynamic point, strings, membranes, and solids. Make the story more complicated by using a trig function:
The same starting and ending points are used. I didn't worry about the symmetry problem since sines are much more complicated that simply adding things up. This time the output value is limited to plus or minus unity due to the sine function.
Since there is only one parameter, the dancing dot is seen. The values for time t and location y are exactly the same, so one gets the straight line in the complex plane. The tx complex plane has a wobble due to the factor of a half. If the domain had been expanded, I am pretty sure one would see a figure 8 in the tx complex plane. This animation required 300 events to look smooth. That will be a problem for more permutations, but forward I went.
The 2-parameter animation has the loops:
I had no idea this would build a tent-like structure. The animation took 90,000 events to create.
The 3-parameter sine animation was too much for my hardware/software. 2.7 million quaternions is too much for my machine. Since I had to stay home on Monday waiting for a new hot water heater, I got a few more shakes of the dice to try and find some combination of parameters that would make a reasonable animation with membranes of sine functions. Here is the best looking result generated so far using 900,000 events:
I needed to delete frames that were blank or only partially filled in. There were only 33 frames in all, but they tell the story. It looks like the wall of the tent structure folding in.
For the 4 parameter case, I tossed 1.2 million quaternions at my old linux box. I got 24 frames. The animation kept blinking from stuff going on the left with that going on the right. To make things look smooth, I rearranged the frames to the stuff on the left happen first, then the stuff on the right. Here is the result:
While I am making no claims that these have physics content at this time, it always amuses me that the quaternion sine function doesn't look like unending sand dunes like I was taught.
This is the pattern seen by increasing the number of parameters in a quaternion animation, so long as a symmetry doesn't play a trick on the results:
|1 parameter||prancing points|
|2 parameters||sliding strings|
|3 parameters||moving membranes|
|4 parameters||slinking solids|
There is a wonderful irony in this table. I am skeptical of the work on strings avoiding the usual label used in the field since it makes the work sound much better than it is in terms of the scientific tradition. Like I tell my three year old, no blood, no band-aid, or in this case, no predictions, no theory. I am also skeptical of my skepticism because I cannot write down a single equation connected with the work going on with strings. What the Donkey Kong are all those way-brighter-than-I-ever-will-be people doing with their efforts? It was interesting that a fad for work on strings is to do "stringie things" in 4D spacetime. At least I can see these dynamic quaternion strings, how the strings merge to form membranes, and how membranes stitch together to form dynamic solids. These visual observations of quaternion animations provide me with a concrete reason to believe work on strings and membranes may have a real value someday, at least if they play in 4D spacetime.
Snarky problem: Imagine what a 10 parameter quaternion expression would look like, then push the knob to 11. Does it make any difference after 4?
Next Monday/Tuesday: Deriving the Einstein field equations in N steps (not sure yet the value of N)