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Mandelbrot Zooms Now Surpass the Scale of the Observable Universe
StartsWithABang writes You're used to real numbers: that is, numbers that can be expressed as a decimal, even if it's an arbitrarily long, non-repeating decimal. There are also complex numbers, which are numbers that have a real part and also an imaginary part. The imaginary part is just like the real part, but is also multiplied by i, or the square root of -1. It's a simple definition: the Mandelbrot set consists of every possible complex number, n, where the sequence n, n^2 + n, (n^2 + n)^2 + n, etc.—where each new term is the prior term, squared, plus n—does not go to either positive or negative infinity. The scale of zoom visualizations now goes well past the limits of the observable Universe, with no signs of loss of complexity at all.
A zoom into a fractal stored as a 16-minute YouTube video must be the least efficient way to store an equation. If only there was some sort of a 'fractal compression' method.
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Incorrect. Abstract mathematical objects are not "encoded within the observable universe"
The set is not encoded in the universe, though the description of the set is. Else, every reference to "infinite" would, well, break the universe.
For large sets, this will be our guide even unto death, for the LORD will work for each type of data it is applied to...
And you think the universe isn't broken NOW? Good god, man. Wake up!
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Mandelbrot Zooms Now Surpass the Scale of the Observable Universe
First off, does that even mean anything? What units is the "scale" of a universe expressed in?
Okay, let's take it to mean the ratio of the size of observable universe to the size of the Planck length, for lack of any better definition. In that case, Mandelzooms surpassed that years ago.
with no signs of loss of complexity at all.
You make it sound like we're expecting a loss of complexity, and we just haven't found it yet. But isn't it mathematically proven that the Mandelbrot set has the same "complexity" at all scales? Kind of inherent in the whole "fractal" thing, I thought...
I'd have thought it would be more interesting to talk about, for example, how all the pretty colours that everyone gawps at aren't even points in the set. They're just colour-coded as to how long the sequence takes to reach a certain value (all of the coloured points ultimately diverge to infinity, which is what makes them not part of the set).
systemd is Roko's Basilisk.
Some of the confusion is that the original description is defined recursively in a way that 'c' only shows up once, and the initial value is not c. z[i] = z[i-1]^2+c. But because z[0] is defined = 0, you can effectively rewrite the sequence in terms of just 'c' starting from the second. The downside is that at first it might LOOK at first glance like the previous term is being added, which is why I like the recursive form.
Also, by not starting from 0 you miss out on a cool connection: for a given fixed C, the graph of convergence for non-zero choices of z[0] over the complex plane gives you a Julia Set. With the neat property that Julia Sets from C inside the Mandelbrot set are fully connected and Julia Sets from C outside the Mandelbrot Set are sparse disconnected Cantor spaces.
"You saved 1968." - Ms. Valerie Pringle to the crew of Apollo 8
persay
That's per se. Go and stand on the naughty step with "peak" guy from the previous post.
Plants are engines powered by the Sun. The very purpose of those leaves is to tap the flow of solar energy. When the giant celestial nuclear reactor is taken into account, the entropy of the entire system is increasing.
Your body is using an external source of energy - the food you eat - to fight the decay.
Forget magic. Any technology distinguishable from divine power is insufficiently advanced.