Can something be deterministic, yet unpredictable? This is the question at the heart of chaos theory. Previously we’ve played the Chaos Game. If I were to give you a set of initial coordinates and a list of functions that are sequentially applied to these coordinates you would probably be able to eventually figure out the final coordinates. But if the functions were chosen at random you would not. Even though the random function is actually not random, but has an underlying deterministic function. In theory, you could, in practice you won’t.

Another thing that would make it impossible to figure out the final coordinates is when you start at an unknown coordinate. To illustrate this: imagine that someone is listing the digits of Pi. When this person starts out and you witness this start you, will be able to follow along and predict the next digit. You could use a deterministic spigot algorithm to do this. The only thing you need to keep track of is the number of digits the person has already listed. Alternatively, you could use a lookup table. If you were really fanatic, however, you would eventually run out of digits and you will have the use the spigot algorithm to get to more digits.

    \[3.14159 \ldots\]

Pi is infinite and transcendental. Every subsequence can occur multiple times. If you were to tune in somewhere in the middle of the person listing its digits you would not be able to predict the next.

    \[\ldots 999999 \ldots\]

What is the next digit in this sequence? This subsequence of six nines occurs for the first time at the 762nd decimal point, also known as the Feynman point. But it also occurs at the 193034th decimal point. Without any context, there’s no way of knowing after which of the infinite subsequences you are supposed to continue.

Given that you’re far enough into the sequence, the digits of Pi can be seen as a pseudo-random number generator. Many things that appear random have underlying deterministic mechanisms. But due to tuning in at an unknown moment, you might not ever figure out what the underlying mechanism is.

The ancient Greeks also thought about this concept. Democritus illustrated the idea with a story of two servants that are sent to get water from a well at the exact same time. The servants would view their meeting as random. Unbeknownst to them, their masters concocted this evil plan. This story would have been a lot better if the masters had the servants walk out to the middle of nowhere from separate routes at the same time. This would have felt a lot more random to them. Of course, this example isn’t totally the same. The servants can easily deduce the underlying reason. The story would be more representative if the servants were mute and did not know sign language or how to write.

“Nothing occurs at random, but everything for a reason and by necessity”

— Leucippus

If you can’t in any practical way make use of the underlying deterministic mechanism, then what’s the use? To be honest I don’t have an answer to this question. I’m still working on one and it’s hard not to fall into a void of existentialism when thinking about it. It doesn’t matter for any practical applications, but it may alter the way we think about the universe. It may lead to searching for hidden mechanisms that we normally wouldn’t look for because we have always treated them as non-deterministic.

“So, you can conceptualise it, but you cannot measure it. Then, does it matter?”

— Dr. Nikolas Sochorakis

Here you find an implementation of the Lorenz System. I don’t understand fluid dynamics, nor will I make an attempt to do. However, I will not view the system as being non-deterministic due to my lack of knowledge. Lorenz’s attractor is fully deterministic, yet has the special property that the same coordinates never occur twice. Convince yourself by trying. This algorithm gave rise to the Chaos movement. I highly recommend reading the book Chaos by James Gleick (also the biographer of Richard Feynman) for a comprehensive overview of the history of this field. The system is very sensitive to initial conditions and I encourage you to try out a bunch of settings. Note that the lines seem aliased/jagged. This is due to the rescaling of the Javascript canvas. For a non-aliased image, you can click on the canvas with your right mouse button and press ‘Save image as’.

Be patient after pressing buy, the image is very large!

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