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this post was submitted on 23 Feb 2024
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No Stupid Questions
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No such thing. Ask away!
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We don't live in an infinitely complex universe as far as we can see. We live on earth almost exclusively and spinning up a solar system is something we can already do.
Given the universe is pretty big and simulations can be simplified (see Heisenberg uncertainty principle etc) it's reasonable that you could spin up a ton of concurrent simulations with only a small step up in needed power usage.
You're also making assumptions about power creation and its finite bounds.
We have already found stuff from other systems entering our own.
The existence of infinite numbers suggests that the power increase required to simulate our universe wouldn't be a small step up in power usage, but an infinite one!
If you assume the only way to simulate an infinite number is storing it in bits, sure. Also, have we ever really done anything to require representing a truly infinite number?
I think you can simulate an infinite number, in a sense, but my issue is whether you can create infinite numbers, even hypothetically, in a simulation.
We simulate Pi all the time, for example. But that simulation of Pi is not Pi. A circle generated by simulated-Pi can only be described with Pi itself, i.e., outside the simulation in a space which does contain infinite numbers.
If you can’t tell the difference, does it matter?
Of course this gets more into Russel’s teapot than occam’s razor territory.
Fractals are infinite
I have to admit that I don't know much about fractals. I have two main questions about this:
Are fractals reaaly infinite? I've heard the coastline of Britain described as fractal, but I'm sure it's not infinite in the sense I understand. As I say, I don't know much about fractals so I may have misunderstood something here.
If fractals are or can be infinite, do computer simulations of fractals actually create fractals of the infinite kind or are they a type of approximation?
Fractal universe theories have been proposed. I don't know many details myself, but just thought it was an example of how you can still have theoretically infinite detail within a finite system.
Technically, a "fractal" is any entity with a fractional dimension. One way to measure this by how its area* multiplies when you scale it up or down. A line that's twice as long has 2x the area. A square twice as wide has 4x the area. A cube has 8x. This implies the formula
scaleFactor = 2^dimension, ordimension = log-base-2(scaleFactor). The Serpinski Triangle is a fractal that contains 3 copies of itself, each at half scale; so if you scale one to be twice as wide, it's equivalent to multiplying the area by 3. From our formula earlier, this means its dimension is log-base-2 of 3, or about 1.585-- somewhere between 1 and 2 dimensional!Note that the Serpinski Triangle is made of copies of itself-- this makes it a "self-similar" fractal, which ironically makes it easier to work with. This is what people generally think of when they say "fractal", and has essentially become the common usage of the term. But note that technically, not all self-similar shapes are fractal (a square can be made of 4 scaled squares), and interestingly, not all fractal shapes are self-similar! Measuring their dimension can be harder, but in your example of eg. the British coastline, notice how the scale at which you measure things changes the length of the coastline. Do you measure each cove? Each tiny protrusion of rock? Each individual grain of sand as the water of the ocean wraps around it? You can compare your answers at different scales and (somehow) use that to calculate a fractional dimension, since they'll scale differently than a flat surface coastline would.
* there's a general name for length/area/volume/etc. which I should be using but I forgot what it is
Edit: Almost forgot to answer your second question; they're an approximation. Computers simulate fractals similarly to how they compute irrational numbers like pi, where they only calculate up to a certain decimal point. For rendering a self-similar fractal, this means they render a certain number of smaller copies, where anything beyond the smallest copy is simply assumed to be in or out of the fractal by default.