Slashdot Mirror
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.
Technically the description of the Mandlebrot set is encoded within the observable universe so there is a problem in recursion her.
Second how is this surprising to anyone ? It's long been possible to describe and mathematically manipulate sets with more elements than the observable universe.
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.
You can't shut us down! The Internet is about the free exchange and sale of other people's ideas!
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
For most complex numbers the sequence will most certainly not converge to positive or negative infinity, whatever those mean. When dealing with complex numbers it only makes sense to talk about a single infinity, which is the point at infinity of the projective complex line (a.k.a. "Riemann sphere").
persay
That's per se. Go and stand on the naughty step with "peak" guy from the previous post.