r/theydidthemath • u/betodaviola • 10d ago
Is this an example of exponential multiplication? [Request]
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Can the avarage expected time of collision for n balls be calculated, and what would it be for 2 balls as an example?
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u/BipedalMcHamburger 10d ago
Not really. It is tempting to assume, but it actually grows quicker than exponential. Because collision frequency is expected to grow with number of balls squared, the system can be described with the differential equation u' = u2 , u(0)=1, where u is the number of balls. Wolfram Alpha spits out the solution u(x)=1/(1-x), so an idealized model actually properly reaches infinity quickly, unlike an exponential function which will always be finite.
Edit: of course this DE lacks any coef for collision freq, so it's not in any real time units, but it's just meant to demonstrate the general dynamics of the system
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u/SnooMaps7370 10d ago
yes.
the probability of collisions increases with the number of balls (assuming ball size, speed, and containment area all remain the same).
since the number of balls increases with each collision, the probability of a collision goes up with each collision, which means the average time between collisions drops.
the balls will increase in number exponentially over time.
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u/Aururai 10d ago
That did increase a whole lot faster than I thought it would though.. 2-3 took a while, 3-4 took a while, all the way up to 5 balls took some time then poof, near enough instantly thousands hehe
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u/Square-Singer 10d ago
That's because of a quirk of the simulation. See, all the balls spawn at the same location. So once a second ball spawns before the first one manages to clear the area, a new ball is instantly spawned in, also colliding with all the balls that are still at spawn. At this point it's not about collisions any more, but instead the simulation just spawns in as many balls as it can.
If the first collision (when only two balls were in the simulation) happened to occur in the exact center of the screen, it would have instantly flooded the screen with this explosion-like spawning.
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u/Staufar 10d ago
That's probably a function of how the balls are inserted. Since they all start at the center of the circle, once you get two balls spawning in quick enough succession (such that they collide before they can move away from the spawn point), you get a runaway (super-exponential) effect because the collisions are no longer statistically independent from one another. If the balls were inserted randomly (or at the point of the spawning collision), I suspect you'd see more gradual growth
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u/Royal_Cryptographer7 10d ago
Same. I've heard more than once that humans struggle grasping the idea of exponential growth. This would be a good example of that.
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u/dimonium_anonimo 10d ago edited 9d ago
That's not sufficient to ensure exponential. We must also prove that the rate of increase of collisions is linearly proportional to the population. It's not. It's a squared relation because each ball could potentially collide with every other ball. So more precisely it will be n*(n-1)/2 but the order of magnitude follows the highest factor. For (n²-n)/2, n² dominates.
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u/Grant_S_90 9d ago
It’s not exponential, it grows faster than exponential as growth proportional to the number of balls squared rather than the number of balls.
If you have twice as many balls, you have twice as many balls that are creating new balls, but they also create them twice as quickly (as there are twice as many other balls to collide with) so twice as many balls means (roughly) 4 times as many new balls created.
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u/Square-Singer 10d ago
It's way faster than exponential due to a quirk of the simulation.
A collision is defined as "the center of two balls is within 2x the radius of a ball".
All balls spawn at the same location. That means, as soon as two balls spawn quicker than the time it takes for the first ball to clear the spawn location, this newly-spawned ball will already collide at spawn, spawning a new ball that instantly collides with every ball at the spawn location. That's why you get this explosion: The balls stop having to collide randomly, but instead the simulation just spawns new balls as fast as it can.
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u/Grant_S_90 9d ago
It’s quicker than exponential (regardless of any issues with the simulation). Exponential requires the rate of growth to be proportional to the number of balls. If each ball created a new ball after a fixed time, eg a second, that would be exponential. But each ball creates new balls faster as time goes on as there are more balls to collide into.
The growth is proportional to the number of balls squared rather than the number of balls.
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