Built with Processing
Built with Processing
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Braidenberg Simulation of Zeno’s Paradox
Achilles vs. the Evil Psychic Tortoise
By Andrew Quitmeyer
Based on: Braitenberg Vehicles Program
Based on program written by William Ngan <contact@metaphorical.net>
And on code by Michael Mateas, Mayhew Seavey, Jason Alderman 2004/05
Revised by Ali Mazalek, Fall 2006
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Click and drag to fire “Flecther’s Paradox” arrows!
Press up on the keyboard if you want to add more Achillesbots (not technically supported since doesn’t make much sense in terms of the original story, and too many achilles makes it go SLOOOOOWWW…but it is fun!)
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In this project I sought to recreate Zeno’s famous paradox within the realm of virtual braitenberg vehicles. Inspiration came from Wallace’s “Everything and More: A Compact History of Infinity” and moreso from the fun, repeated use of the Tortoise and Achilles characters in Hofstadter’s “Godel, Escher, Bach.” In case the reader is not familiar with Zeno’s paradox, the idea runs mostly as follows:
Achilles, the famous greek hero, was one day challenged to a race by a clever tortoise. The tortoise stipulated that if he was only given a small head start, he would be sure to win the race. The tortoise went on to logically “prove” the outcome of the race by stating that no matter how much faster achilles may be than the tortoise, by the time that achilles reached the starting position fo the tortoise, the tortoise will also have moved some distance, and by the time that achilles reaches the tortoise’s new position, the tortoise will have moved once more. The idea is that no matter how infintesimal the distance moved by the tortoise, since he had a head start he will remain in the lead. Achille’s, no match for the wit of the tortoise, then concedes the match to the seemingly inevitable winner, the Tortoise.
This paradox is often rephrased in the following manner:
If Achilles wants to reach the finish line, he must first move half the distance to the goal. After that, he would have to traverse half of the remaining distance... and half of the distance after that- ad infinitum. Consequently, Achilles would approach the goal but never be able to actually reach it. Obviously there is something weird about this logical puzzle since we have all been able to arrive at places instead of just infinitely approaching locations. In fact, if this logic were true it would disprove motion altogether since, before Achilles could even reach the halfway point, he would have to make it halfway between the starting line and the halfway point, and before he could get there he would need to make it halfway between the start and the ¼ mark…
Difficulties in simulation:
The two largest difficulties encountered in trying to simulate this contest of crazy logic versus common sense were 1) designing a program to realistically function yet depict an impossible, purely mental behavior, and 2) performance issues with the base Braitenberg code.
1. Creating the paradox:
It was really hard to think of a depiction of the paradox that would also be interesting. Since the paradox technically disproves motion one way to simulate it would be to have all the characters remain motionless, and Achilles say, “You win” to the tortoise. Not too interesting. Alternately we could have Achilles just go straight for the goal but stop infinitesimally close to it. Again, boring. In order for this project to be interesting at all there were going to have to be times that Achilles did not obey this crazy logic along with times that he did obey it. Thus I derived the idea of the evil, psychic tortoise with the ability of implanting these crazy thoughts in the impressionable Achilles head, but the range at which he could implant these thoughts was limited. Therefore there would be a gradient constructed with which Achilles behavior was driven. If he got too close to the tortoise his confusion would max out and he would be almost entirely driven by the tortoise’s logic, and if the tortoise were far away Achilles would behave with his normal, common sense.
This is accomplished by having 3 interconnected, sensory fields, one representing the real goal, one displaying mental goals implanted by the tortoise, and one that implements the tortoise’s psychic influence. Here’s how it works:
Both Achilles and the Tortoise are modded, loverbots which seek out bright sources, with the addition that they have the addition of a constant translational velocity.
The tortoise, unhindered by his own crazy logic, constantly seeks and moves toward the true goal, while broadcasting crazy waves of influence. Achilles will constantly seek out and move toward the true goal at a speed 10 times greater than the tortoise (preserving the 10:1 ratio of the original paradox) unless he falls within the tortoise’s mind rays. In that case the tortoise will place a faux goal on the ground located halfway between Achilles and the true goal (he whispers to him mentally, “Well if you want to get to that goal, first you have to go halfway!” ). The intensity of these pseudo-goals is created by how much influence the tortoise had over Achilles at the time of the fake goal’s creation. Thus, if the tortoise is far away, the fake goal may seem dim in comparison to the real goal, but if Achilles is being bombarded by this mental radiation the intensity of the fake source (and Achilles’s confusion) will be so great that he knows, no matter what, before he reaches the real goal he definitely must go halfway!
To spice things up even more, some other features were added:
Perhaps you are sick of that tortoise lording it over Achilles and you want to mess with the outcome of the simulation. Well you can click and drag your mouse to fire arrows from the related “Fletcher’s Paradox” (not true Fletcher’s Paradox arrows since that deals with the infinite division of time as opposed to space). If these arrows hit the tortoise right his mental powers will be temporarily diminished (his score is protected by his thick shell), but the arrows are also affected by the tortoise’s mental abilities (they are modified explorer bots) and the arrows have to be right on target or they will slow down as the tortoise proves their motion impossible and eventually deflect.
Achilles is super strong even if his mind is weak, so he can go very fast and shove the tortoise around if he needs to. He does have one other weakness however (let’s call it his “Achilles heel” if you will). If Achilles gets hit in a tiny, tiny area near his back (around his heel?) he will lose all the points he had gotten thus far and be temporarily physically disabled.
The program is also set up to add more Achilles by pressing “UP” on the keyboard. This feature isn’t technically supported since it violates the original story and if you add too many Achilles, it drastically reduces the performance.
2. Performance issues
The original Braitenberg code is not really meant to have constantly moving light sources so other efforts were made to keep the program going at a good clip. I think it actually runs pretty super well even with up to 8 or so Achilles battling it out at the same time.
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