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.

Comments

Re:Ehhh What ?

[disputationist]

Incorrect. Abstract mathematical objects are not "encoded within the observable universe"

Re:Ehhh What ?

[X0563511]

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.

It's that twat with the upside down head again.

[Hognoxious]

It's not n^2 + n, it's n^2 + c.

That's to say, the number you multiply by itself isn't the same as the number you add.

Re:It's that twat with the upside down head again.

[Mateorabi]

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.

Positive or negative infinity?

[PacoSuarez]

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").

Re:confused

[wonkey_monkey]

The idea is that the "scale" of the observable universe is the ratio from the largest "thing" (the whole observable universe) to the smallest "thing," which is the Planck length. That ratio is 10^63 or something like that, much less than the zoom level that's achieved in the video.